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Remember that $4x$ means $4\\times x$.","highlights":[{"text":"Complete the table of values for $y = 4x - 4$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y=4\\htmlData{anchor=xneg2}{\\color{@sky}{(-2)}}-4$"},{"latex":"$$y=\\htmlData{anchor=yneg12}{\\color{@green}{-12}}$"},{"latex":"$$y=4\\htmlData{anchor=xneg1}{\\color{@sky}{(-1)}}-4$"},{"latex":"$$y=\\htmlData{anchor=yneg8}{\\color{@green}{-8}}$"},{"latex":"$$y=4\\htmlData{anchor=xzero}{\\color{@sky}{(0)}}-4$"},{"latex":"$$y=\\htmlData{anchor=yneg4}{\\color{@green}{-4}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"xneg2"},{"color":"sky","style":"text-color","anchor":"xneg1"},{"color":"sky","style":"text-color","anchor":"xzero"},{"color":"green","style":"text-color-with-background","anchor":"yneg12"}]},"order":1,"title":"Calculate non-positive inputs","description":"Substitute each non-positive input into the formula. Brackets keep a negative input together."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y=4\\htmlData{anchor=xone}{\\color{@sky}{(1)}}-4$"},{"latex":"$$y=\\htmlData{anchor=yzero}{\\color{@green}{0}}$"},{"latex":"$$y=4\\htmlData{anchor=xtwo}{\\color{@sky}{(2)}}-4$"},{"latex":"$$y=\\htmlData{anchor=yfour}{\\color{@green}{4}}$"},{"latex":"$$y=4\\htmlData{anchor=xthree}{\\color{@sky}{(3)}}-4$"},{"latex":"$$y=\\htmlData{anchor=yeight}{\\color{@green}{8}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"xone"},{"color":"sky","style":"text-color","anchor":"xtwo"},{"color":"sky","style":"text-color","anchor":"xthree"},{"color":"green","style":"text-color-with-background","anchor":"yeight"}]},"order":2,"title":"Calculate positive inputs","description":"Substitute each positive input into the formula in the same way."},{"type":"text","order":3,"title":"Complete the table","content":"Enter the results beneath their matching $x$-values:\n\n| $x$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$ |\n|---|---:|---:|---:|---:|---:|---:|\n| $y$ | $-12$ | $-8$ | $-4$ | $0$ | $4$ | $8$ |\n\nAll six correct entries earn **2 marks**. Two or three correct entries earn **1 mark**."}]}],"generatedAt":"2026-07-19T16:36:45.133Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"c44090e1-6445-46c9-bc6a-2b3f011aece1","label":"b","content":"On the grid, draw the graph of $y = 4x - 4$ for values of $x$ from -2 to 3\n\n![A coordinate grid for drawing the graph of y = 4x - 4.](https://assets.mathsgenie.co.uk/booklet/a82548ceb78980cb45f817d805c5e676adf26d16a17c56d50b1aa1d99bc939bf.jpg)","markscheme":"- Points plotted correctly from table M1\n- Correct straight line drawn from $x = -2$ to $x = 3$ A1\n\n![A coordinate grid for drawing y = 4x − 4, with x-values from −2 to 3 and y-values from −12 to 8.](https://assets.mathsgenie.co.uk/question-figures/v1/9e38c158e6ef86f321148944479ea189.svg)","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Plots at least 2 points correctly from their table.","code":"M"},{"id":"p2","marks":1,"description":"Draws a correct straight line through all points for the range $x = -2$ to $x = 3$.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-05T10:32:41.417Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"table_and_line","label":"Make a table and join","steps":[{"type":"text","order":0,"title":"Choose the inputs","content":"Use the whole-number $x$-values across the required interval: $-2$, $-1$, $0$, $1$, $2$ and $3$. For each one, substitute into $y=4x-4$.","highlights":[{"text":"for values of $x$ from -2 to 3","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y=4x-4$"},{"latex":"$$\\begin{array}{c|rrrrrr}x&-2&-1&0&1&2&3\\\\ \\hline y&\\htmlData{anchor=outputs}{\\color{@sky}{-12}}&\\color{@sky}{-8}&\\color{@sky}{-4}&\\color{@sky}{0}&\\color{@sky}{4}&\\color{@sky}{8}\\end{array}$"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"outputs"}]},"order":1,"title":"Calculate the outputs","description":"Multiply each input by $4$, then subtract $4$. A value table keeps the coordinates organised."},{"type":"graph","order":2,"title":"Plot the points","viewport":{"top":9.5,"left":-2.6,"right":3.6,"bottom":-13.5},"axisLabels":{"x":"x","y":"y"},"description":"Plot each ordered pair from the table. These correctly plotted points earn M1.","expressions":[{"id":"expr-0","color":"sky","label":"(-2, -12)","latex":"(-2,-12)","pointStyle":"point","labelOrientation":"right"},{"id":"expr-1","color":"sky","label":"(-1, -8)","latex":"(-1,-8)","pointStyle":"point","labelOrientation":"right"},{"id":"expr-2","color":"sky","label":"(0, -4)","latex":"(0,-4)","pointStyle":"point","labelOrientation":"right"},{"id":"expr-3","color":"sky","label":"(1, 0)","latex":"(1,0)","pointStyle":"point","labelOrientation":"above"},{"id":"expr-4","color":"sky","label":"(2, 4)","latex":"(2,4)","pointStyle":"point","labelOrientation":"left"},{"id":"expr-5","color":"sky","label":"(3, 8)","latex":"(3,8)","pointStyle":"point","labelOrientation":"left"}]},{"type":"graph","order":3,"title":"Join with a ruler","viewport":{"top":9.5,"left":-2.6,"right":3.6,"bottom":-13.5},"axisLabels":{"x":"x","y":"y"},"description":"Use a ruler to draw one straight line through the points, stopping at the required endpoints $(-2,-12)$ and $(3,8)$. The correct segment earns A1.","expressions":[{"id":"expr-0","color":"green","latex":"y=(4x-4)\\left\\{-2\\le x\\le3\\right\\}","lineStyle":"solid"},{"id":"expr-1","color":"green","label":"(-2, -12)","latex":"(-2,-12)","pointStyle":"point","labelOrientation":"right"},{"id":"expr-2","color":"sky","latex":"(-1,-8)","pointStyle":"point"},{"id":"expr-3","color":"sky","latex":"(0,-4)","pointStyle":"point"},{"id":"expr-4","color":"sky","latex":"(1,0)","pointStyle":"point"},{"id":"expr-5","color":"sky","latex":"(2,4)","pointStyle":"point"},{"id":"expr-6","color":"green","label":"(3, 8)","latex":"(3,8)","pointStyle":"point","labelOrientation":"left"},{"id":"expr-7","color":"green","label":"y = 4x - 4","latex":"(2.4,5.6)","hidden":true,"labelOrientation":"above"}]}]}],"generatedAt":"2026-07-19T16:40:21.628Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":38},"walkthroughVideo":null},{"id":"146f04b9-0f2e-4a5d-b4e9-16ef2e6537eb","revisionId":"8664f9b5-77f3-4f32-82c1-329f3331ac15","questionNumber":null,"questionType":"LA","specification":"","difficultyBand":"easy","tier":null,"totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":false,"parts":[{"id":"4e260623-256f-4f65-8d8d-9629595480c0","label":"a","content":"On the grid, draw the graph of $y = 2x + 5$ for values of $x$ from -3 to 2.\n\n![A coordinate grid for drawing the graph of y = 2x + 5.](https://assets.mathsgenie.co.uk/figures/coords/844938bf1bf8c0316c3ad9357a97e726011aa0d2fd0ba6be2b0c71964fb105b3.png)","markscheme":"- Correct table of values or at least 2 points correctly calculated B1\n- Points plotted correctly M1\n- Correct straight line drawn from $x = -3$ to $x = 2$ A1\n\n![Solution](https://assets.mathsgenie.co.uk/figures/coords/5c848a66c468776a7cdd62cce0b6e16fc06460414a000b5e5d824b55678e4ab6.png)","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Correct table of values or at least 2 points correctly calculated (e.g., (-3, -1), (0, 5), (2, 9)).","code":"B"},{"id":"p2","marks":1,"description":"At least 2 points from their table or the correct equation plotted correctly.","code":"M"},{"id":"p3","marks":1,"description":"Correct straight line drawn from x = -3 to x = 2.","code":"A"}],"finalAnswers":["Line passing through (-3, -1) and (2, 9)"],"generatedAt":"2026-05-06T13:59:52.634Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"plot_straight_line","label":"Plot the straight line","steps":[{"type":"text","order":0,"title":"Make a value table","content":"Use the given $x$-values from $-3$ to $2$. For each one, substitute into $y=2x+5$ and record the result:\n\n$$\n\\begin{array}{c|rrrrrr}\nx&-3&-2&-1&0&1&2\\\\\n\\hline\ny&-1&1&3&5&7&9\n\\end{array}\n$$\n\nThis earns the **B1** mark: a correct table, or at least two correctly calculated points, is enough for this mark.","highlights":[{"text":"for values of $x$ from -3 to 2","location":"specification"}]},{"type":"graph","order":1,"title":"Plot the points","viewport":{"top":10,"left":-4,"right":4,"bottom":-2},"axisLabels":{"x":"x","y":"y"},"description":"Plot each ordered pair from the table. The first coordinate gives the horizontal position and the second gives the vertical position. Correctly plotting the points earns the M1 mark.","expressions":[{"id":"expr-0","color":"sky","label":"(-3, -1)","latex":"(-3,-1)","pointStyle":"point","labelOrientation":"below"},{"id":"expr-1","color":"sky","label":"(-2, 1)","latex":"(-2,1)","pointStyle":"point","labelOrientation":"above"},{"id":"expr-2","color":"sky","label":"(-1, 3)","latex":"(-1,3)","pointStyle":"point","labelOrientation":"left"},{"id":"expr-3","color":"sky","label":"(0, 5)","latex":"(0,5)","pointStyle":"point","labelOrientation":"right"},{"id":"expr-4","color":"sky","label":"(1, 7)","latex":"(1,7)","pointStyle":"point","labelOrientation":"right"},{"id":"expr-5","color":"sky","label":"(2, 9)","latex":"(2,9)","pointStyle":"point","labelOrientation":"right"}]},{"type":"graph","order":2,"title":"Join with a ruler","viewport":{"top":10,"left":-4,"right":4,"bottom":-2},"axisLabels":{"x":"x","y":"y"},"description":"Use a ruler to draw one straight line through the plotted points. Stop at the endpoints where x = -3 and x = 2; do not extend the line across the whole grid. This correct line segment earns the A1 mark.","expressions":[{"id":"expr-0","color":"sky","label":"x=-3","latex":"(-3,-1)","pointStyle":"point","labelOrientation":"below"},{"id":"expr-1","color":"sky","latex":"(-2,1)","pointStyle":"point"},{"id":"expr-2","color":"sky","latex":"(-1,3)","pointStyle":"point"},{"id":"expr-3","color":"sky","latex":"(0,5)","pointStyle":"point"},{"id":"expr-4","color":"sky","latex":"(1,7)","pointStyle":"point"},{"id":"expr-5","color":"sky","label":"x=2","latex":"(2,9)","pointStyle":"point","labelOrientation":"above"},{"id":"expr-6","color":"green","latex":"y=2x+5\\left\\{-3\\le x\\le2\\right\\}"},{"id":"expr-7","color":"green","label":"y = 2x + 5","latex":"(1.5,8)","hidden":true,"labelOrientation":"left"}]}]}],"generatedAt":"2026-07-19T18:09:27.464Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"4e260623-256f-4f65-8d8d-9629595480c0:table_of_values","label":"Using a Table of Values","steps":[{"type":"text","order":0,"title":"Set up a table","content":"To draw the graph of the linear equation $$y = 2x + 5$$ for $x$-values from $-3$ to $2$, we will first set up a table of values. This will give us the coordinates we need to plot on the grid.\n\nWe choose integer values of $x$ from $-3$ to $2$:\n\n$$\\begin{array}{|c|c|c|c|c|c|c|} \\hline x & -3 & -2 & -1 & 0 & 1 & 2 \\\\ \\hline y & & & & & & \\\\ \\hline \\end{array}$$","highlights":[{"text":"y = 2x + 5","location":"specification"},{"text":"values of x from -3 to 2","location":"specification"}]},{"type":"text","order":1,"title":"Calculate the y-values","content":"Now, we substitute each $x$-value into the equation $$y = 2x + 5$$ to find its corresponding $y$-value:\n\n* For $x = -3$: \n $$y = 2(-3) + 5 = -6 + 5 = -1$$\n* For $x = -2$: \n $$y = 2(-2) + 5 = -4 + 5 = 1$$\n* For $x = -1$: \n $$y = 2(-1) + 5 = -2 + 5 = 3$$\n* For $x = 0$: \n $$y = 2(0) + 5 = 0 + 5 = 5$$\n* For $x = 1$: \n $$y = 2(1) + 5 = 2 + 5 = 7$$\n* For $x = 2$: \n $$y = 2(2) + 5 = 4 + 5 = 9$$\n\nOur completed table of values is:\n\n$$\\begin{array}{|c|c|c|c|c|c|c|} \\hline x & -3 & -2 & -1 & 0 & 1 & 2 \\\\ \\hline y & -1 & 1 & 3 & 5 & 7 & 9 \\\\ \\hline \\end{array}$$"},{"type":"text","order":2,"title":"Plot the points","content":"Next, write down the coordinates from the table as pairs $(x, y)$:\n\n$$\\begin{aligned} &(-3, -1) \\\\ &(-2, 1) \\\\ &(-1, 3) \\\\ &(0, 5) \\\\ &(1, 7) \\\\ &(2, 9) \\end{aligned}$$\n\nCarefully plot each of these points on the provided coordinate grid. Make sure they lie in a straight line, as $y = 2x + 5$ is a linear graph."},{"type":"text","order":3,"title":"Draw the line","content":"Use a ruler to draw a single, straight line passing through all your plotted points. \n\nEnsure your line goes all the way from $x = -3$ to $x = 2$ to fully satisfy the range requested in the question.\n\n### Markscheme Breakdown:\n* **B1**: Finding at least two correct points or drawing a correct table of values.\n* **M1**: Plotting the calculated coordinates correctly on the axes.\n* **A1**: Drawing a fully correct, continuous straight line from $x = -3$ to $x = 2$."}]},{"id":"4e260623-256f-4f65-8d8d-9629595480c0:gradient_intercept","label":"Using Gradient and Y-Intercept","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\t{Compare } y = \\htmlData{anchor=m-val}{\\color{@sky}{2}}x + \\htmlData{anchor=c-val}{\\color{@amber}{5}} \\text{ with the general line equation:}$"},{"latex":"$$y = \\htmlData{anchor=m-var}{\\color{@sky}{m}}x + \\htmlData{anchor=c-var}{\\color{@amber}{c}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"m-val"},{"color":"amber","style":"text-color","anchor":"c-val"}],"connections":[{"to":"m-var","from":"m-val","color":"sky","label":"gradient","style":"solid"},{"to":"c-var","from":"c-val","color":"amber","label":"y-intercept","style":"solid"}]},"order":0,"title":"Identify gradient and intercept","description":"By comparing the equation to $y = mx + c$, we can find both the $y$-intercept ($c$) and the gradient ($m$) of the line."},{"type":"text","order":1,"title":"Plot the y-intercept","content":"The $y$-intercept is $c = 5$, which corresponds to the coordinate point:\n\n$$(0, 5)$$\n\nStart by plotting this point on your grid directly on the vertical $y$-axis."},{"type":"text","order":2,"title":"Apply the gradient","content":"The gradient is $m = 2$. This tells us the slope of the line:\n\n$$\\text{Gradient} = \\frac{\\text{Change in } y}{\\text{Change in } x} = \\frac{2}{1}$$\n\nThis means for every $1$ unit we move to the right (positive $x$-direction), we must move up by $2$ units (positive $y$-direction).\n\nLet's apply this to find more points starting from $(0, 5)$:\n* Move $1$ right, $2$ up $\\rightarrow (1, 7)$\n* Move another $1$ right, $2$ up $\\rightarrow (2, 9)$\n\nWe can also move to the left by doing the opposite (move $1$ left, $2$ down):\n* Move $1$ left, $2$ down $\\rightarrow (-1, 3)$\n* Move another $1$ left, $2$ down $\\rightarrow (-2, 1)$\n* Move one more left, $2$ down $\\rightarrow (-3, -1)$"},{"type":"text","order":3,"title":"Draw the straight line","content":"Plot all of these points on your grid:\n\n$$(-3, -1), \\quad (-2, 1), \\quad (-1, 3), \\quad (0, 5), \\quad (1, 7), \\quad (2, 9)$$\n\nUsing a ruler, draw a single straight line passing through all of these points. Ensure your line goes exactly from $x = -3$ to $x = 2$ as specified in the question.","highlights":[{"text":"draw the graph of $y = 2x + 5$ for values of $x$ from -3 to 2.","location":"specification"}]}]},{"id":"4e260623-256f-4f65-8d8d-9629595480c0:endpoint_calculation","label":"Plotting Endpoints","steps":[{"type":"text","order":0,"title":"Identify the endpoints","content":"To draw a straight-line graph, we only need to find the coordinates of its two outermost points (the endpoints of our interval) and connect them with a straight line.\n\nThe question asks us to draw the graph of the linear equation $$y = 2x + 5$$ for values of $x$ from $-3$ to $2$.\n\nSo, our boundary values for $x$ are:\n- The left endpoint: $x = -3$\n- The right endpoint: $x = 2$","highlights":[{"text":"y = 2x + 5","location":"specification"},{"text":"from -3 to 2","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\text{Substitute } \title{x} = \text{\\color{@sky}{-3}} \text{ into the equation } y = 2x + 5:$"},{"latex":"$$y = 2(\\htmlData{anchor=x-val1}{\\color{@sky}{-3}}) + 5$"},{"latex":"$$y = \\htmlData{anchor=calc1}{\\color{@pink}{-6}} + 5$"},{"latex":"$$y = \\htmlData{anchor=y-val1}{\\color{@pink}{-1}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"x-val1"},{"color":"pink","style":"text-color","anchor":"y-val1"}],"connections":[{"to":"calc1","from":"x-val1","color":"pink","label":"multiply by 2","style":"solid"},{"to":"y-val1","from":"calc1","color":"pink","label":"add 5","style":"solid"}]},"order":1,"title":"Calculate the left endpoint","description":"This gives us our first coordinate point on the grid: $(-3, -1)$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\text{Substitute } \title{x} = \text{\\color{@amber}{2}} \text{ into the equation } y = 2x + 5:$"},{"latex":"$$y = 2(\\htmlData{anchor=x-val2}{\\color{@amber}{2}}) + 5$"},{"latex":"$$y = \\htmlData{anchor=calc2}{\\color{@purple}{4}} + 5$"},{"latex":"$$y = \\htmlData{anchor=y-val2}{\\color{@purple}{9}}$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"x-val2"},{"color":"purple","style":"text-color","anchor":"y-val2"}],"connections":[{"to":"calc2","from":"x-val2","color":"purple","label":"multiply by 2","style":"solid"},{"to":"y-val2","from":"calc2","color":"purple","label":"add 5","style":"solid"}]},"order":2,"title":"Calculate the right endpoint","description":"This gives us our second coordinate point on the grid: $(2, 9)$."},{"type":"text","order":3,"title":"Plot points and draw","content":"Follow these final steps on your grid to complete the graph:\n\n1. **Plot the first endpoint**: Find $x = -3$ on the horizontal axis and move down to $y = -1$. Mark this point $(-3, -1)$ with a small cross $\\times$ or dot.\n2. **Plot the second endpoint**: Find $x = 2$ on the horizontal axis and move up to $y = 9$. Mark this point $(2, 9)$ with a small cross $\\times$ or dot.\n3. **Draw the line**: Place a ruler aligning these two plotted points and draw a single, clean straight line connecting them.\n\n*Tip: Ensure your line starts exactly at $x = -3$ and ends exactly at $x = 2$ to fully cover the required range!*"}]}],"generatedAt":"2026-06-12T10:26:46.517Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":55},"walkthroughVideo":null},{"id":"32f38b56-b569-426e-bb82-609aa610db6f","revisionId":"e5d043cc-86f5-49a0-9f38-b0b50436fa93","questionNumber":null,"questionType":"LA","specification":"","difficultyBand":"easy","tier":null,"totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":false,"parts":[{"id":"81016829-2f40-4822-b061-5e9a7109cd03","label":"a","content":"On the grid, draw the graph of $y = 2x + 3$ for values of $x$ from -3 to 3\n\n![A coordinate grid for drawing the graph of y = 2x + 3.](https://assets.mathsgenie.co.uk/figures/coords/1fe01197fa53bd8555b607cd4d5635ecd7e748710ac80b7bf231e7112cb82059.png)","markscheme":"- Correct table of values or at least 2 points correctly calculated B1\n- Points plotted correctly M1\n- Correct straight line drawn from $x = -3$ to $x = 3$ A1\n\n![Solution](https://assets.mathsgenie.co.uk/figures/coords/3d5394317e4a9546a51fc273ba8a78bf43a8ac4f1a9e429c7759cc5498a79d79.png)","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Correct table of values or at least 2 points correctly calculated (e.g., (-3, -3), (0, 3), (3, 9))","code":"B"},{"id":"p2","marks":1,"description":"At least 2 points plotted correctly from their table or calculations","code":"M"},{"id":"p3","marks":1,"description":"A fully correct straight line drawn from x = -3 to x = 3","code":"A"}],"finalAnswers":["y = 2x + 3"],"generatedAt":"2026-05-06T14:00:05.717Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"table_and_plot","label":"Table of values and plotting","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y=2(-3)+3=\\htmlData{anchor=yneg}{-3}$"},{"latex":"$$(-3,\\htmlData{anchor=pneg}{-3})$"},{"latex":"$$y=2(0)+3=\\htmlData{anchor=yzero}{3}$"},{"latex":"$$(0,\\htmlData{anchor=pzero}{3})$"},{"latex":"$$y=2(3)+3=\\htmlData{anchor=ypos}{9}$"},{"latex":"$$(3,\\htmlData{anchor=ppos}{9})$"}],"connections":[{"to":"pneg","from":"yneg","color":"sky","label":"use as the y-coordinate"},{"to":"pzero","from":"yzero","color":"green","label":"use as the y-coordinate"},{"to":"ppos","from":"ypos","color":"orange","label":"use as the y-coordinate"}]},"order":0,"title":"Calculate three points","description":"Substitute convenient values across the required interval into the rule. These correctly calculated points earn the B1 mark."},{"type":"graph","order":1,"title":"Plot the points","viewport":{"top":10,"left":-4,"right":4,"bottom":-4},"axisLabels":{"x":"x","y":"y"},"description":"Plot each coordinate accurately on the grid. Correct plotting earns the M1 mark.","expressions":[{"id":"expr-0","color":"sky","label":"(-3, -3)","latex":"(-3,-3)","pointStyle":"point","labelOrientation":"above"},{"id":"expr-1","color":"green","label":"(0, 3)","latex":"(0,3)","pointStyle":"point","labelOrientation":"right"},{"id":"expr-2","color":"amber","label":"(3, 9)","latex":"(3,9)","pointStyle":"point","labelOrientation":"below"}]},{"type":"graph","order":2,"title":"Join with a ruler","viewport":{"top":10,"left":-4,"right":4,"bottom":-4},"axisLabels":{"x":"x","y":"y"},"description":"Use a ruler to draw one straight line through the points, stopping at the required endpoints where x = -3 and x = 3. The correct straight line over this whole interval earns the A1 mark.","expressions":[{"id":"expr-0","color":"indigo","latex":"y=2x+3\\left\\{-3\\le x\\le3\\right\\}","lineStyle":"solid"},{"id":"expr-1","color":"sky","label":"(-3, -3)","latex":"(-3,-3)","pointStyle":"point","labelOrientation":"above"},{"id":"expr-2","color":"green","label":"(0, 3)","latex":"(0,3)","pointStyle":"point","labelOrientation":"right"},{"id":"expr-3","color":"amber","label":"(3, 9)","latex":"(3,9)","pointStyle":"point","labelOrientation":"below"},{"id":"expr-4","color":"indigo","label":"y = 2x + 3","latex":"(2.5,8)","hidden":true,"labelOrientation":"left"}]}]}],"generatedAt":"2026-07-20T10:44:49.478Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"81016829-2f40-4822-b061-5e9a7109cd03:tableofvalues","label":"Using a Table of Values","steps":[{"type":"text","order":0,"title":"Set up a table","content":"To draw the graph of the equation $$y = 2x + 3$$ we need to find the corresponding $y$-coordinates for several $x$-values between $-3$ and $3$. Let's choose the integer values of $x$ from $-3$ to $3$ and set up a table of values:\n\n$$\\begin{array}{|c|r|r|r|r|r|r|r|} \\hline x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\\\ \\hline y & & & & & & & \\\\ \\hline \\end{array}$$","highlights":[{"text":"draw the graph of y = 2x + 3 for values of x from -3 to 3","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{For } \\htmlData{anchor=x-val1}{\\color{@sky}{x = -3}}: \\quad y = 2(\\color{@sky}{-3}) + 3 = -6 + 3 = \\htmlData{anchor=y-val1}{\\color{@pink}{-3}}$"},{"latex":"$$\\text{For } \\htmlData{anchor=x-val2}{\\color{@amber}{x = 0}}: \\quad y = 2(\\color{@amber}{0}) + 3 = 0 + 3 = \\htmlData{anchor=y-val2}{\\color{@purple}{3}}$"},{"latex":"$$\\text{For } \\htmlData{anchor=x-val3}{\\color{@green}{x = 3}}: \\quad y = 2(\\color{@green}{3}) + 3 = 6 + 3 = \\htmlData{anchor=y-val3}{\\color{@teal}{9}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"x-val1"},{"color":"pink","style":"text-color","anchor":"y-val1"},{"color":"amber","style":"text-color","anchor":"x-val2"},{"color":"purple","style":"text-color","anchor":"y-val2"},{"color":"green","style":"text-color","anchor":"x-val3"},{"color":"teal","style":"text-color","anchor":"y-val3"}],"connections":[{"to":"y-val1","from":"x-val1","color":"pink","label":"substitute","style":"solid"},{"to":"y-val2","from":"x-val2","color":"purple","label":"substitute","style":"solid"},{"to":"y-val3","from":"x-val3","color":"teal","label":"substitute","style":"solid"}]},"order":1,"title":"Calculate the y-values","description":"Substitute the $x$-values into the equation $y = 2x + 3$ to find the corresponding $y$-values."},{"type":"text","order":2,"title":"Complete the table","content":"By calculating the $y$-value for each integer $x$, we obtain the completed table of values:\n\n$$\\begin{array}{|c|r|r|r|r|r|r|r|} \\hline x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\\\ \\hline y & -3 & -1 & 1 & 3 & 5 & 7 & 9 \\\\ \\hline \\end{array}$$\n\nEach column in our table gives us a coordinate pair $(x, y)$ that lies on the line:\n* $(-3, -3)$\n* $(-2, -1)$\n* $(-1, 1)$\n* $(0, 3)$\n* $(1, 5)$\n* $(2, 7)$\n* $(3, 9)$\n\nCompleting this table or calculating at least two of these points correctly earns you your first mark **[1 mark]**."},{"type":"text","order":3,"title":"Use calculator to verify","content":"You can easily generate this table of values using your calculator to save time and double-check your manual calculations.","calculator":[{"gdc":{"expression":"y = 2x + 3","instructions":"1. Go to the **Table** menu.\n2. Input the formula `y = 2x + 3`.\n3. Set the range for $x$ starting at `-3`, ending at `3`, with a step of `1`.\n4. View the table of values."},"ti84":{"expression":"Y1 = 2X + 3","instructions":"1. Press [MODE] and select **TABLE** (if available) or press [Y=].\n2. Enter `2X + 3` into `\nY1`.\n3. Press [2nd] [WINDOW] (TBLSET) and set `TblStart = -3` and `ΔTbl = 1`.\n4. Press [2nd] [GRAPH] (TABLE) to view the table of values."},"desmos":{"expression":"y_1 = 2x_1 + 3","instructions":"1. Click the **+** button at the top left of the sidebar and select **Table**.\n2. In the header of the first column, enter `x_1`.\n3. In the header of the second column, enter `2x_1 + 3`.\n4. Type values from `-3` to `3` in the $x_1$ column to automatically generate the $y$-values."},"description":"Generate a table of values for $y = 2x + 3$"}]},{"type":"text","order":4,"title":"Plot the points","content":"Now, plot each coordinate pair onto the grid:\n\n1. Move along the horizontal $x$-axis to find the $x$-coordinate.\n2. Move up or down to find the $y$-coordinate.\n3. Mark the position with a neat dot or a small cross 'x'.\n\nEnsure that all points are plotted accurately. Accurate plotting of these points earns the second mark **[1 mark]**."},{"type":"text","order":5,"title":"Draw the line","content":"Using a ruler, draw a single, straight line that goes through all of your plotted points.\n\n* **CRITICAL:** Make sure your line goes all the way from $x = -3$ to $x = 3$.\n\nOnce the line is drawn through all the correct points from $x = -3$ to $x = 3$, you secure the final mark **[1 mark]**."}]},{"id":"81016829-2f40-4822-b061-5e9a7109cd03:gradientintercept","label":"Using Gradient and Y-Intercept","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y = \\htmlData{anchor=gradient}{\\color{@sky}{2}}x + \\htmlData{anchor=intercept}{\\color{@amber}{3}}$"},{"latex":"$$\\text{Y-intercept at } (0, \\htmlData{anchor=y-coord}{\\color{@amber}{3}})$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"intercept"},{"color":"amber","style":"text-color","anchor":"y-coord"}],"connections":[{"to":"y-coord","from":"intercept","color":"amber","label":"y-intercept","style":"solid"}]},"order":0,"title":"Identify the Y-Intercept","description":"An equation of a straight line is written in the form $y = mx + c$, where $m$ is the gradient and $c$ is the $y$-intercept. For the equation $y = 2x + 3$, the constant term is $+3$, which means the graph crosses the $y$-axis at $(0, 3)$."},{"type":"text","order":1,"title":"Use the Gradient","content":"The gradient of the line is $2$. This means that for every $1$ unit we move to the right along the $x$-axis, we move up by $2$ units along the $y$-axis:\n\n$$\\text{Gradient} = \\frac{\\text{Change in } y}{\\text{Change in } x} = \\frac{2}{1}$$\n\nStarting from our $y$-intercept at $(0, 3)$:\n- Move right $1$ unit and up $2$ units to plot $(1, 5)$.\n- Move right another $1$ unit and up $2$ units to plot $(2, 7)$.\n- Move left $1$ unit and down $2$ units to plot $(-1, 1)$, and so on."},{"type":"hyper_latex","block":{"nodes":[{"text":"For the upper boundary:"},{"latex":"$$x = \\htmlData{anchor=x-upper}{\\color{@pink}{3}} \\implies y = 2(\\color{@pink}{3}) + 3 = \\htmlData{anchor=y-upper}{\\color{@purple}{9}}$"},{"text":"For the lower boundary:"},{"latex":"$$x = \\htmlData{anchor=x-lower}{\\color{@teal}{-3}} \\implies y = 2(\\color{@teal}{-3}) + 3 = \\htmlData{anchor=y-lower}{\\color{@red}{-3}}$"}],"highlights":[{"color":"pink","style":"text-color","anchor":"x-upper"},{"color":"purple","style":"text-color","anchor":"y-upper"},{"color":"teal","style":"text-color","anchor":"x-lower"},{"color":"red","style":"text-color","anchor":"y-lower"}],"connections":[{"to":"y-upper","from":"x-upper","color":"pink","label":"Substitute x = 3","style":"dashed"},{"to":"y-lower","from":"x-lower","color":"teal","label":"Substitute x = -3","style":"dashed"}]},"order":2,"title":"Determine the Endpoints","description":"We need to draw the graph specifically for $x$ values from $-3$ to $3$. We can check our endpoints to make sure our line spans the full range required by the question."},{"type":"text","order":3,"title":"Plot and Connect","content":"Now we plot the points we found on the grid:\n- $(-3, -3)$\n- $(0, 3)$\n- $(3, 9)$\n\nUsing a ruler, draw a single, continuous straight line passing through these points, starting exactly at $x = -3$ and ending at $x = 3$.\n\n*Note on Marks:* \n- You get **1 mark** (B1) for correctly calculating at least 2 points.\n- You get **1 mark** (M1) for plotting those points accurately on the grid.\n- You get the final **1 mark** (A1) for drawing the straight line from $x = -3$ to $x = 3$.","calculator":[{"gdc":{"expression":"y = 2x + 3","instructions":"1. Open the Graph menu and enter 2x + 3.\n2. Open the Table tool to check the y-values for x = -3, 0, and 3."},"ti84":{"expression":"Y1 = 2X + 3","instructions":"1. Press [Y=] and enter 2X + 3 next to Y1.\n2. Press [2nd] then [GRAPH] (TABLE) to view the coordinate values for each integer X from -3 to 3."},"desmos":{"expression":"y = 2x + 3 \\{-3 \\le x \\le 3\\}","instructions":"1. Type y = 2x + 3 in the first input bar.\n2. Click the gear icon and select 'Convert to table' to view the coordinates from x = -3 to 3."},"description":"You can verify the coordinates using a graphing calculator."}]}]}],"generatedAt":"2026-06-12T13:35:04.383Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":76},"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/review/6f2d6853-cc6e-4172-96c7-e6b4b44773de.mp4","durationSeconds":84,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/review/6f2d6853-cc6e-4172-96c7-e6b4b44773de.jpg"}}],"subjectCode":"gcse_maths__wjec","summaries":[{"id":"ed0aa6e6-7eca-4813-8fa3-4eaa899e51fc","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"146f04b9-0f2e-4a5d-b4e9-16ef2e6537eb","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"32f38b56-b569-426e-bb82-609aa610db6f","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"5b64a292-f3bc-4705-b27d-fa47ee4c895e","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"6b48c40a-1846-4e29-93af-909edd75d314","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"7d637825-43d1-4126-8b09-dbf53555d8ee","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"939a3eb8-84fb-41ae-b06b-f052759acde5","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"c398326d-d937-4500-b28f-744a330da057","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"eb97b467-0f4f-4010-a963-45b91c26515f","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"0134009f-49ea-441d-b56a-b46600816d62","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"2522da3c-350e-44e8-9e61-223627158528","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"84550a5b-91a4-4e7c-8614-8d1490af3583","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"b93bf801-786c-444f-8e31-fb543b9c7bd1","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"c2e9d857-8335-4e2d-9b9a-ec1305ec204c","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"ca2a3e1f-4ec6-4c43-888a-be2c9b581ce1","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"f7fb7b7c-a5f8-4efb-8c36-fcb76a0d51d8","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"d37511de-168d-44c3-8ed5-6cb203f91f77","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"13688535-664c-493a-92dc-826716346764","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"7229cab2-a3f5-4d31-bb6a-fa6b19e431be","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"9dc5f3ab-7d9a-4a10-9413-ec5d55184f36","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"b65edc97-d6a4-4cc6-886f-d5b0dc306038","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"c4f10484-831d-420c-b40a-cd4fb98d50ec","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"cb6ae992-8d3f-4b40-b76b-0b90aa4af311","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"cede7162-805d-46ce-b982-d9934f2bce05","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"2d296b62-9588-4e75-a464-ac80ddd20515","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"d2b96531-d36f-468e-8250-50c02df95fc1","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"d681bd52-e934-4b6d-9c08-9186a00e6570","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"ecaf5372-9646-4d22-bf8e-2ee8d685b027","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false}],"title":"Drawing Linear Graphs","topicId":"43b3ff0e-f970-fc63-7b68-b72092e4b3a4","topicSidebar":"$L3a","topicSlug":"grade-3-3-15-drawing-linear-graphs","topicTitle":"Drawing Linear Graphs"}]}],"$L3b","$L3c"]}]