^2$ to confirm they represent the same curve."},"ti84":{"instructions":"1. Press [MODE], arrow down to 'FUNCTION' and change it to 'PARAMETRIC' (PAR).\n2. Press [Y=] and enter:\n X1T = T - 1\n Y1T = 6 - T^2\n3. Change MODE back to 'FUNCTION' (FUNC).\n4. Press [Y=] and enter:\n Y1 = 6 - (X + 1)^2\n5. Press [GRAPH] and verify both graphs match exactly."},"desmos":{"expression":"(t - 1, 6 - t^2)","instructions":"1. Enter the parametric curve as a coordinate pair: (t - 1, 6 - t^2).\n2. On a new line, enter the Cartesian equation: y = 6 - (x + 1)^2.\n3. Observe that both plots overlap exactly."},"description":"You can verify that this Cartesian equation is correct by graphing both the parametric equations and your Cartesian equation to see if they overlap perfectly."}]}]}],"generatedAt":"2026-06-09T07:02:46.376Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"abc3a820-1afd-497b-a415-1f90cdff0190","label":"b","content":"Sketch the curve","markscheme":"- Correct shape of an inverted parabola (n-shape) B1\n- Maximum point correctly labeled at $(-1, 6)$ B1\n- $y$-intercept correctly labeled at $5$ B1","marks":3,"order":1,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Inverted (n-shaped) parabola.","code":"B"},{"id":"p2","marks":1,"description":"Correct maximum point at $(-1, 6)$.","code":"B"},{"id":"p3","marks":1,"description":"Correct $y$-intercept at $5$.","code":"B"}],"finalAnswers":[],"generatedAt":"2026-05-06T14:04:23.293Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"cartesian_conversion","label":"Conversion to Cartesian Form","steps":[{"type":"text","order":0,"title":"Rearrange for parameter $t$","content":"To find the Cartesian equation of the curve, we first express the parameter $t$ in terms of $x$ using the $x$ equation:\n\n$$x = t - 1$$\n\nAdding $1$ to both sides gives:\n\n$$t = x + 1$$","highlights":[{"text":"x = t - 1","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$t = \\color{@sky}{x + 1}$"},{"latex":"$$y = 6 - \\htmlData{anchor=t-term}{t}^2$"},{"latex":"$$y = 6 - \\htmlData{anchor=t-sub}{\\color{@sky}{(x + 1)}}^2$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"t-term"},{"color":"sky","style":"text-color","anchor":"t-sub"}],"connections":[{"to":"t-sub","from":"t-term","color":"sky","label":"substitute","style":"solid"}]},"order":1,"title":"Substitute into $y$","description":"Now, we substitute this expression for $t$ into the equation for $y$ to eliminate the parameter."},{"type":"text","order":2,"title":"Identify key features","content":"Our Cartesian equation is in vertex form:\n\n$$y = -(x + 1)^2 + 6$$\n\nFrom this, we can easily identify the key features needed to sketch the curve:\n\n1. **Shape**: The coefficient of the $x^2$ term is negative (due to the minus sign in front of the bracket), which means we have an **inverted (n-shaped) parabola** B1.\n\n2. **Vertex (Maximum Point)**: The maximum value of $y$ occurs when the squared term $(x+1)^2 = 0$, which is at $x = -1$. Substituting $x = -1$ gives $y = 6$. So, the maximum point is at **$(-1, 6)$** B1.\n\n3. **$y$-intercept**: Set $x = 0$ to find where the curve crosses the $y$-axis:\n\n$$\\begin{aligned} y &= 6 - (0 + 1)^2 \\\\ y &= 6 - 1 \\\\ y &= 5 \\end{aligned}$$\n\nSo, the $y$-intercept is at **$(0, 5)$** B1."},{"type":"text","order":3,"title":"Sketch the curve","content":"Using the features identified, we can now draw the sketch:\n\n- Draw a set of coordinate axes.\n- Plot the maximum vertex at $(-1, 6)$.\n- Plot the $y$-intercept at $(0, 5)$.\n- Draw a smooth, symmetrical inverted parabolic curve passing through these points.","calculator":[{"gdc":{"expression":"x(t) = t - 1, y(t) = 6 - t^2","instructions":"1. Open the graphing application and switch the graph type to 'Parametric'.\n2. Enter the equations: x(t) = t - 1 and y(t) = 6 - t^2.\n3. Adjust the window settings to display the vertex at (-1, 6) and y-inter

1.6 C: Coordinate geometry in the (x, y) plane

Question 20

1.6 C: Coordinate geometry in the (x, y) plane Questions

1.6 C: Coordinate geometry in the (x, y) plane

Question 20

1.6 C: Coordinate geometry in the (x, y) plane Questions