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The product of two negative numbers is positive."},{"text":"$$-9$","correct":false,"explanation":"This comes from applying the square only to $3$ and then retaining the negative sign. The whole value $-3$ is squared, so $(-3)^2 = 9$."},{"text":"$$6$","correct":false,"explanation":"This treats squaring as multiplying the number by $2$. Squaring means multiplying the number by itself, so $(-3) \\times (-3) = 9$."},{"text":"$$-6$","correct":false,"explanation":"This combines doubling with incorrectly retaining the negative sign. The operation is squaring, not doubling, and $(-3)^2$ equals $9$."}]},{"id":"28c0335c-7c02-26b7-b8e2-167c88a5417e","cardType":"cloze","points":null,"front":"When squaring a negative number, always use [ ] to ensure the result is positive.","back":"When squaring a negative number, always use brackets to ensure the result is positive.","mcqOptions":[{"text":"brackets","correct":true,"explanation":"Brackets show that the negative sign is part of the number being squared. For example, $(-3)^2 = (-3) \\times (-3) = 9$."},{"text":"minus signs","correct":false,"explanation":"A minus sign alone does not ensure the entire negative number is squared. Brackets are required: $(-3)^2 = 9$, whereas $-3^2$ is interpreted as $-(3^2)$."},{"text":"square roots","correct":false,"explanation":"Square roots reverse squaring rather than showing that a negative number is being squared. Brackets enclose the whole negative number, as in $(-3)^2 = 9$."},{"text":"decimal points","correct":false,"explanation":"Decimal points indicate place value and do not control which parts of an expression are squared. Brackets correctly group the negative number, giving $(-3)^2 = 9$."}]},{"id":"209c4188-8c0d-4348-0d5a-7d8db680099f","cardType":"basic","points":1,"front":"What is the specific name for the smooth curve formed by a quadratic graph?","back":"Parabola","mcqOptions":[{"text":"Parabola","correct":true,"explanation":"The graph of a quadratic function is called a parabola. It is a smooth, symmetrical curve that may open upwards or downwards."},{"text":"Hyperbola","correct":false,"explanation":"A hyperbola is associated with relationships such as inverse proportion, not ordinary quadratic functions. The smooth curve formed by a quadratic graph is called a parabola."},{"text":"Ellipse","correct":false,"explanation":"An ellipse is a closed oval-shaped curve and is not the graph of a quadratic function in the form $y = ax^2 + bx + c$. That graph is a parabola."},{"text":"Straight line","correct":false,"explanation":"A straight-line graph represents a linear relationship with no $x^2$ term. A quadratic relationship produces a curved graph specifically called a parabola."}]},{"id":"bf2509ec-af48-5223-3882-1cf0d4d61d6b","cardType":"cloze","points":null,"front":"A quadratic graph opens upwards (like a smile) if the $x^2$ coefficient is [ ].","back":"A quadratic graph opens upwards (like a smile) if the $x^2$ coefficient is positive.","mcqOptions":[{"text":"positive","correct":true,"explanation":"When the coefficient of $x^2$ is positive, increasing $|x|$ makes the quadratic term positive, so the parabola opens upwards like a smile."},{"text":"negative","correct":false,"explanation":"This reverses the relationship between the coefficient’s sign and the graph’s direction. A negative $x^2$ coefficient makes the parabola open downwards; a positive coefficient makes it open upwards."},{"text":"zero","correct":false,"explanation":"If the $x^2$ coefficient is zero, the expression is not quadratic and does not form a parabola. An upward-opening quadratic requires a positive $x^2$ coefficient."},{"text":"squared","correct":false,"explanation":"The variable $x$ is squared, but the coefficient itself need not be a square number. Its sign determines direction: a positive coefficient makes the graph open upwards."}]},{"id":"56de27b1-17f8-9ee0-6351-26620cd644b7","cardType":"cloze","points":null,"front":"A quadratic graph opens downwards (like a frown) if the $x^2$ coefficient is [ ].","back":"A quadratic graph opens downwards (like a frown) if the $x^2$ coefficient is negative.","mcqOptions":[{"text":"negative","correct":true,"explanation":"A negative coefficient of $x^2$ makes the quadratic term decrease as $|x|$ increases, so the parabola opens downwards like a frown."},{"text":"positive","correct":false,"explanation":"This reverses the rule governing the parabola’s direction. A positive $x^2$ coefficient produces an upward-opening graph, while a negative coefficient produces a downward-opening graph."},{"text":"zero","correct":false,"explanation":"A zero $x^2$ coefficient removes the quadratic term, so the graph is no longer a parabola. A downward-opening quadratic must have a negative $x^2$ coefficient."},{"text":"squared","correct":false,"explanation":"The fact that $x$ is squared does not decide whether the graph opens upwards or downwards. The coefficient must specifically be negative for it to open downwards."}]},{"id":"93314cbc-36c7-ef92-34ce-c96f63b4ad96","cardType":"basic","points":1,"front":"How should you join the plotted points when drawing a quadratic graph?","back":"With one smooth curve (do not use a ruler).","mcqOptions":[{"text":"With one smooth curve (do not use a ruler).","correct":true,"explanation":"A quadratic changes gradient continuously, so its plotted points should be connected by one smooth curve. Straight ruler segments would incorrectly introduce sharp changes in gradient."},{"text":"With one straight line (use a ruler).","correct":false,"explanation":"A straight line represents a linear relationship and cannot show the changing gradient of a quadratic. The points should be joined with one smooth curve without using a ruler."},{"text":"With separate unjoined points (do not draw a curve).","correct":false,"explanation":"The plotted points are samples from a continuous quadratic relationship, so leaving them unjoined does not show the complete graph. They should be connected with one smooth curve."},{"text":"With several sharp corners (join each point exactly).","correct":false,"explanation":"Joining points with sharp corners creates piecewise straight sections and sudden gradient changes that a quadratic does not have. The correct graph uses one continuous smooth curve."}]},{"id":"6de2c6a9-95f9-26f0-5541-743cb500ee94","cardType":"basic","points":1,"front":"What is the \"turning point\" of a quadratic graph?","back":"The lowest or highest point of the parabola.","mcqOptions":[{"text":"The lowest or highest point of the parabola.","correct":true,"explanation":"The turning point is where the parabola changes direction. It is the minimum point for an upward-opening parabola or the maximum point for a downward-opening parabola."},{"text":"The leftmost or rightmost point of the parabola.","correct":false,"explanation":"A standard quadratic parabola continues indefinitely to the left and right, so it has no leftmost or rightmost point. Its turning point is its lowest or highest point."},{"text":"The first or second crossing of the x-axis.","correct":false,"explanation":"Crossings of the x-axis are roots, not turning points. The turning point is where the parabola changes direction and reaches its lowest or highest value."},{"text":"The lowest or highest crossing of the y-axis.","correct":false,"explanation":"A quadratic crosses the y-axis only once, and that intercept is not generally an extreme value. The turning point is the parabola’s lowest or highest point."}]},{"id":"cab6b37b-65a0-2509-9577-9dbc8f8b73b4","cardType":"basic","points":1,"front":"How are the \"roots\" of a quadratic identified from its graph?","back":"They are the $x$-values where the curve crosses the x-axis.","mcqOptions":[{"text":"They are the $x$-values where the curve crosses the x-axis.","correct":true,"explanation":"On the x-axis, $y = 0$, so the corresponding $x$-coordinates solve the quadratic equation. These x-axis intersections are therefore the roots of the quadratic."},{"text":"They are the $y$-values where the curve crosses the x-axis.","correct":false,"explanation":"Every point on the x-axis has $y = 0$, so the y-values do not distinguish the roots. The roots are the corresponding $x$-values where the curve crosses the x-axis."},{"text":"They are the $x$-values where the curve crosses the y-axis.","correct":false,"explanation":"The y-axis has $x = 0$, so its intersection gives the y-intercept rather than the roots. Roots are the $x$-values where the curve crosses the x-axis."},{"text":"They are the $x$-values where the curve reaches its turning point.","correct":false,"explanation":"The turning point identifies the parabola’s maximum or minimum, not generally where $y = 0$. Roots are the $x$-values at which the graph crosses the x-axis."}]},{"id":"79c888e1-aad4-4e73-6e4b-8d25e23b74c8","cardType":"basic","points":1,"front":"At any root on a quadratic graph, what is the value of $y$?","back":"$$0$","mcqOptions":[{"text":"$$0$","correct":true,"explanation":"A root is where the quadratic curve crosses or touches the $x$-axis. Every point on the $x$-axis has a $y$-coordinate of $0$."},{"text":"$$1$","correct":false,"explanation":"This confuses a root with a point one unit above the $x$-axis. At a root, the curve lies on the $x$-axis, so $y=0$."},{"text":"$$-1$","correct":false,"explanation":"A negative $y$-value places the point below the $x$-axis rather than at a root. Roots occur where the curve has $y=0$."},{"text":"$$x$","correct":false,"explanation":"This confuses the variable coordinate with its value at an intercept. A root can have various $x$-coordinates, but its $y$-coordinate is always $0$."}]},{"id":"fa175b1c-635f-2759-cc73-c2f4dd398a49","cardType":"cloze","points":null,"front":"In a table of values for a quadratic, the $y$-values usually form a [ ] or a consistent curve.","back":"In a table of values for a quadratic, the $y$-values usually form a symmetrical pattern or a consistent curve.","mcqOptions":[{"text":"symmetrical pattern","correct":true,"explanation":"Quadratic graphs are symmetrical about their vertical axis, so equally spaced $x$-values on opposite sides of that axis produce matching $y$-values."},{"text":"linear pattern","correct":false,"explanation":"A linear pattern has constant first differences and represents a straight line. Quadratic values instead show symmetry and usually have constant second differences."},{"text":"random pattern","correct":false,"explanation":"Quadratic values are determined by a fixed equation, so they are not random. They form a symmetrical pattern around the graph’s vertical axis."},{"text":"alternating pattern","correct":false,"explanation":"Alternating values repeatedly switch according to position, which is not a general property of quadratics. Their $y$-values form a symmetrical pattern around the axis."}]},{"id":"43b0de44-fb0a-5332-6de6-3ba77f228f4a","cardType":"basic","points":2,"front":"How do you solve $x^2 - x - 4 = -2$ using the graph of $y = x^2 - x - 4$?","back":"Find the $x$-coordinates where the curve intersects the horizontal line $y = -2$.","mcqOptions":[{"text":"Find the $x$-coordinates where the curve intersects the horizontal line $y = -2$.","correct":true,"explanation":"The curve represents $y=x^2-x-4$, so solutions to the expression equalling $-2$ occur where the curve reaches $y=-2$."},{"text":"Find the $x$-coordinates where the curve intersects the horizontal line $y = 0$.","correct":false,"explanation":"Using $y=0$ would find the roots of $x^2-x-4=0$. Here the expression equals $-2$, so the required horizontal line is $y=-2$."},{"text":"Find the $x$-coordinates where the curve intersects the horizontal line $y = 2$.","correct":false,"explanation":"This loses the negative sign on the right-hand side and would solve $x^2-x-4=2$. The given equation requires intersections with $y=-2$."},{"text":"Find the $x$-coordinates where the curve intersects the vertical line $x = -2$.","correct":false,"explanation":"A vertical line fixes an $x$-value rather than locating where the quadratic has a specified output. Draw the horizontal line $y=-2$ and read the intersection $x$-coordinates."}]},{"id":"2d907676-f61a-6aa3-2659-40ba167461ac","cardType":"basic","points":2,"front":"What must be done to the equation $x^2 = 3x + 4$ before solving it using the roots of a graph?","back":"Rearrange it to $x^2 - 3x - 4 = 0$ so one side equals zero.","mcqOptions":[{"text":"Rearrange it to $x^2 - 3x - 4 = 0$ so one side equals zero.","correct":true,"explanation":"Subtracting $3x+4$ from both sides gives $x^2-3x-4=0$. The graph’s roots then provide the solutions because roots occur when $y=0$."},{"text":"Rearrange it to $x^2 - 3x + 4 = 0$ so one side equals zero.","correct":false,"explanation":"The constant changes incorrectly here. Subtracting $4$ from both sides produces $-4$, so the correct rearrangement is $x^2-3x-4=0$."},{"text":"Rearrange it to $x^2 + 3x - 4 = 0$ so one side equals zero.","correct":false,"explanation":"Moving $3x$ from the right to the left changes its sign to negative, not positive. The required equation is $x^2-3x-4=0$."},{"text":"Rearrange it to $x^2 - 3x - 3 = 0$ so one side equals zero.","correct":false,"explanation":"This introduces an off-by-one error in the constant term. Subtracting the original $4$ gives $x^2-3x-4=0$, whose roots solve the equation."}]},{"id":"2570afd2-ead6-717b-2a53-a15b2943a869","cardType":"cloze","points":null,"front":"To solve an equation where the quadratic expression equals a number, you must draw or imagine a [ ] at that $y$-value.","back":"To solve an equation where the quadratic expression equals a number, you must draw or imagine a horizontal line at that $y$-value.","mcqOptions":[{"text":"horizontal line","correct":true,"explanation":"A fixed $y$-value is represented by a horizontal line. Its intersections with the quadratic identify the $x$-values for which the expression equals that number."},{"text":"vertical line","correct":false,"explanation":"A vertical line represents a fixed $x$-value, not a fixed output. To solve where the expression equals a number, use a horizontal line at that $y$-value."},{"text":"tangent line","correct":false,"explanation":"A tangent only touches the curve locally and is not generally used to represent a specified output. A fixed $y$-value requires a horizontal line."},{"text":"diagonal line","correct":false,"explanation":"A diagonal line makes $y$ vary with $x$, so it does not represent one fixed value. The correct construction is a horizontal line at the required $y$-value."}]},{"id":"637ed387-4665-e1b4-5639-c727477258a9","cardType":"basic","points":2,"front":"For the graph $y = x^2 - 2x - 1$, calculate $y$ when $x = -1$.","back":"$$2$ (calculated as $1 + 2 - 1$)","mcqOptions":[{"text":"$$2$ (calculated as $1 + 2 - 1$)","correct":true,"explanation":"Substituting $x=-1$ gives $(-1)^2=1$ and $-2(-1)=2$. Therefore, $1+2-1=2$."},{"text":"$$-2$ (calculated as $1 - 2 - 1$)","correct":false,"explanation":"This misses that multiplying two negative values gives a positive value: $-2(-1)=2$, not $-2$. The correct calculation is $1+2-1=2$."},{"text":"$$4$ (calculated as $1 + 2 + 1$)","correct":false,"explanation":"This changes the final constant from $-1$ to $+1$. Keeping its original sign gives $1+2-1=2$."},{"text":"$$-4$ (calculated as $-1 - 2 - 1$)","correct":false,"explanation":"This mishandles both $(-1)^2$, which equals $1$, and $-2(-1)$, which equals $2$. The correct result is $1+2-1=2$."}]},{"id":"c6f7d3df-2b37-68a1-a546-e4320a7513c0","cardType":"basic","points":1,"front":"What is the coordinate of the turning point for $y = x^2 - 4x + 3$ if the lowest point is at $x = 2$ and $y = -1$?","back":"$$(2, -1)$","mcqOptions":[{"text":"$$(2, -1)$","correct":true,"explanation":"Coordinates are written in the order $(x,y)$. With the lowest point at $x=2$ and $y=-1$, the turning point is $(2,-1)$."},{"text":"$$(-1, 2)$","correct":false,"explanation":"This reverses the coordinate order. Since $x=2$ and $y=-1$ at the turning point, the coordinate must be written as $(2,-1)$."},{"text":"$$(2, 1)$","correct":false,"explanation":"This loses the negative sign of the turning point’s $y$-value. The stated lowest point has $x=2$ and $y=-1$, giving $(2,-1)$."},{"text":"$$(-2, -1)$","correct":false,"explanation":"This incorrectly changes the sign of the $x$-coordinate. The turning point is stated to occur at $x=2$, so its coordinate is $(2,-1)$."}]},{"id":"25e8657f-4ef7-6e83-85f5-c5326ddb2e7f","cardType":"cloze","points":null,"front":"If a quadratic graph has a negative $x^2$ term, the turning point will be the [ ] point on the curve.","back":"If a quadratic graph has a negative $x^2$ term, the turning point will be the highest point on the curve.","mcqOptions":[{"text":"highest","correct":true,"explanation":"A negative coefficient of $x^2$ makes the parabola open downwards. Its turning point is therefore a maximum, meaning it is the highest point on the curve."},{"text":"lowest","correct":false,"explanation":"The lowest turning point occurs when the parabola opens upwards, usually with a positive $x^2$ coefficient. A negative coefficient makes the turning point the highest point."},{"text":"leftmost","correct":false,"explanation":"The sign of the $x^2$ coefficient controls whether the curve opens upwards or downwards, not its horizontal extreme. A negative coefficient gives a highest point."},{"text":"rightmost","correct":false,"explanation":"A standard vertical parabola continues indefinitely in horizontal directions, so its turning point is not a rightmost point. With a negative $x^2$ coefficient, it is the highest point."}]},{"id":"a03554e7-3fea-6aa1-551b-62e7945591e9","cardType":"basic","points":1,"front":"In the coordinate $(x, y)$, which direction does the $x$ value represent?","back":"The horizontal position (across).","mcqOptions":[{"text":"The horizontal position (across).","correct":true,"explanation":"The $x$-coordinate shows how far left or right a point lies from the origin, so it represents the horizontal position."},{"text":"The vertical position (upwards).","correct":false,"explanation":"This confuses the two coordinate axes. The $y$-coordinate gives the vertical position, while the $x$-coordinate gives the horizontal position across the graph."},{"text":"The diagonal position (across).","correct":false,"explanation":"Coordinates do not directly specify a diagonal position. The $x$ value represents horizontal movement across the graph, while the $y$ value represents vertical movement."},{"text":"The vertical position (downwards).","correct":false,"explanation":"The direction of the $y$-axis has been confused with the role of the $x$-axis. The $x$ value represents the horizontal position across the graph."}]},{"id":"a65929cc-2fb6-dcb7-9aea-8609e8fc3d53","cardType":"cloze","points":null,"front":"If a value in your table of values looks wildly out of place compared to the curve, you should [ ] it.","back":"If a value in your table of values looks wildly out of place compared to the curve, you should recalculate it.","mcqOptions":[{"text":"recalculate","correct":true,"explanation":"An isolated value that does not follow the curve may result from an arithmetic or substitution error, so calculating it again checks and corrects the result."},{"text":"estimate","correct":false,"explanation":"Replacing the value with an estimate could hide the original calculation error. You should recalculate it carefully using the corresponding $x$ value and the quadratic equation."},{"text":"ignore","correct":false,"explanation":"Ignoring an unusual result may leave an incorrect point in the table or graph. You should recalculate it to check for a substitution or arithmetic error."},{"text":"plot","correct":false,"explanation":"Plotting the suspicious value without checking it may distort the quadratic curve. You should recalculate it first to confirm whether the table entry is accurate."}]},{"id":"080152e1-8a3d-6f9a-8b17-deadfbbc0246","cardType":"basic","points":2,"front":"What are the two solutions to $x^2 - x - 4 = 2$ if the graph crosses $y = 2$ at $x = -2$ and $x = 3$?","back":"

$x = -2$
$x = 3$

","mcqOptions":[{"text":"

$x = -2$
$x = 3$

","correct":true,"explanation":"Solutions at a specified $y$ value are the $x$-coordinates of the intersections. The graph meets $y=2$ at $x=-2$ and $x=3$."},{"text":"

$x = -3$
$x = 2$

","correct":false,"explanation":"This reverses the signs associated with the two factors. Solving $x^2-x-6=0$ gives $(x-3)(x+2)=0$, so $x=-2$ and $x=3$."},{"text":"

$x = -2$
$x = 2$

","correct":false,"explanation":"This assumes the intersections are symmetrical about the $y$-axis. The quadratic’s axis of symmetry is $x=0.5$, so the solutions are $x=-2$ and $x=3$."},{"text":"

$x = 2$
$x = 3$

","correct":false,"explanation":"This mistakes the horizontal line’s value, $y=2$, for one of the $x$-coordinates. The actual intersection coordinates give $x=-2$ and $x=3$."}]},{"id":"7acbffb2-2f4d-addb-c344-1623fb2ef124","cardType":"basic","points":1,"front":"Why is $y = 7x - x^2$ a \"frown\" shaped graph?","back":"The coefficient of the $x^2$ term is negative.","mcqOptions":[{"text":"The coefficient of the $x^2$ term is negative.","correct":true,"explanation":"A negative coefficient of $x^2$ makes a quadratic open downwards, producing an upside-down U or “frown” shape with a maximum point."},{"text":"The coefficient of the $x^2$ term is positive.","correct":false,"explanation":"A positive coefficient would make the parabola open upwards in a “smile” shape. Here the coefficient is $-1$, so the graph opens downwards."},{"text":"The coefficient of the $x^2$ term is zero.","correct":false,"explanation":"A zero coefficient would remove the quadratic term entirely, producing a straight-line equation. The coefficient is $-1$, which makes the parabola open downwards."},{"text":"The coefficient of the $x^2$ term is 7.","correct":false,"explanation":"This confuses the coefficient of $x$ with the coefficient of $x^2$. The quadratic coefficient is $-1$, and its negative sign creates the frown shape."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Paper 1"},"heading":"Drawing Quadratic Graphs","title":"Drawing Quadratic Graphs Flashcards","topbar":{"backHref":"/gcse/maths/ocr/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"16a686d5-67c6-418b-ba57-66ae95971e55","walled":"$L39"}],"$L3a","$L3b"]