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It opens upwards or downwards depending on the coefficient of $x^2$."},{"text":"Hyperbola","correct":false,"explanation":"A hyperbola consists of separate curved branches and is commonly associated with reciprocal functions. The smooth U-shaped curve of a quadratic graph is called a parabola."},{"text":"Straight line","correct":false,"explanation":"A straight line represents a linear relationship with a constant gradient. A quadratic relationship instead produces a smooth U-shaped curve called a parabola."},{"text":"Circle","correct":false,"explanation":"A circle is a closed curve whose points are equally distant from its centre. The open U-shaped curve formed by a quadratic graph is a parabola."}]},{"id":"d2071746-b351-4c0f-a662-fbc72be2e116","cardType":"basic","points":1,"front":"How can you identify a quadratic equation from its terms?","back":"The highest power of $x$ is $x^2$.","mcqOptions":[{"text":"The highest power of $x$ is $x^2$.","correct":true,"explanation":"A quadratic equation has degree 2, meaning the greatest exponent of $x$ is 2. Its highest-power term therefore contains $x^2$."},{"text":"The highest power of $x$ is $x$.","correct":false,"explanation":"An equation whose highest power is $x$ has degree 1 and is linear, not quadratic. A quadratic equation has $x^2$ as its highest power."},{"text":"The highest power of $x$ is $x^3$.","correct":false,"explanation":"An equation with $x^3$ as its highest power has degree 3 and is cubic. A quadratic equation is degree 2, so its highest power is $x^2$."},{"text":"The highest power of $x$ is $x^0$.","correct":false,"explanation":"An equation containing only $x^0$, or constants, has no variable term and is not quadratic. A quadratic equation must have $x^2$ as its highest power."}]},{"id":"6fd7d6e9-58b9-4b00-a4d9-4e314d12fdc9","cardType":"basic","points":1,"front":"Why does substituting $x = -3$ into $y = x^2$ result in a positive 9?","back":"A negative number squared always becomes positive: $(-3)^2 = 9$.","mcqOptions":[{"text":"A negative number squared always becomes positive: $(-3)^2 = 9$.","correct":true,"explanation":"Squaring means multiplying the number by itself. Since the product of two negative numbers is positive, $(-3)(-3)=9$."},{"text":"A negative number squared always remains negative: $(-3)^2 = -9$.","correct":false,"explanation":"Keeping the result negative overlooks that squaring multiplies two negative factors. Their signs cancel, so $(-3)^2=(-3)(-3)=9$, which is positive."},{"text":"A negative number squared always becomes doubled: $(-3)^2 = -6$.","correct":false,"explanation":"Multiplying the exponent by the number confuses squaring with doubling. Squaring requires multiplication by itself, so $(-3)^2=(-3)(-3)=9$."},{"text":"A negative number squared always becomes positive: $(-3)^2 = 3$.","correct":false,"explanation":"Changing only the sign gives the additive inverse rather than the square. Squaring means multiplying $-3$ by itself, producing the positive value 9."}]},{"id":"27c11319-ff74-48ec-851c-c6fc70c6a468","cardType":"basic","points":2,"front":"Calculate the y-value for $y = x^2 - 2x - 3$ when $x = -2$.","back":"

$(-2)^2 - 2(-2) - 3$
$4 + 4 - 3 = 5$

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

$(-2)^2 - 2(-2) - 3$
$4 + 4 - 3 = 5$

","correct":true,"explanation":"Substituting $x=-2$ gives $(-2)^2=4$ and $-2(-2)=4$. Combining the terms gives $4+4-3=5$."},{"text":"

$(-2)^2 - 2(-2) - 3$
$4 - 4 - 3 = -3$

","correct":false,"explanation":"The middle term has been given the wrong sign. Since $-2(-2)=+4$, the calculation should be $4+4-3=5$."},{"text":"

$(-2)^2 - 2(-2) - 3$
$-4 + 4 - 3 = -3$

","correct":false,"explanation":"Treating $(-2)^2$ as $-4$ ignores that two negative factors multiply to a positive. The correct calculation is $4+4-3=5$."},{"text":"

$(-2)^2 - 2(-2) - 3$
$4 + 4 + 3 = 11$

","correct":false,"explanation":"The constant term is $-3$, not $+3$. Keeping its negative sign gives $4+4-3=5$, rather than adding all three values."}]},{"id":"aea5db4f-acc9-42be-968b-f19f858df62e","cardType":"cloze","points":null,"front":"When substituting negative values into squared terms, you should always use [ ] to avoid errors.","back":"When substituting negative values into squared terms, you should always use brackets to avoid errors.","mcqOptions":[{"text":"brackets","correct":true,"explanation":"Brackets keep the negative sign attached to the substituted value, so an expression such as $(-3)^2$ is interpreted correctly as 9."},{"text":"square roots","correct":false,"explanation":"Square roots reverse squaring in suitable contexts but do not clarify substitution. Brackets are needed to keep a negative value together, as in $(-3)^2$."},{"text":"decimals","correct":false,"explanation":"Decimals change how a number is written but do not prevent sign errors when squaring. Brackets correctly show that the whole negative value is squared."},{"text":"fractions","correct":false,"explanation":"Fractions are a form of number and do not protect a negative substitution from sign mistakes. Brackets should enclose the negative value before it is squared."}]},{"id":"7b916f34-39c3-4d56-893c-43168b9e36d8","cardType":"basic","points":1,"front":"What pattern do y-values typically show in a quadratic table of values?","back":"Symmetry around the turning point.","mcqOptions":[{"text":"Symmetry around the turning point.","correct":true,"explanation":"Quadratic graphs are mirror-symmetric about the vertical line through their turning point. Consequently, x-values equally far from this line produce equal y-values."},{"text":"Constant increase around the turning point.","correct":false,"explanation":"A constant increase in y-values is characteristic of a linear relationship, not a quadratic one. Quadratic y-values instead show symmetry around the turning point."},{"text":"Random variation around the turning point.","correct":false,"explanation":"Random variation would not reflect the structured pattern of a quadratic function. Its y-values are symmetric for x-values equally spaced on either side of the turning point."},{"text":"Constant decrease around the turning point.","correct":false,"explanation":"A constant decrease describes a linear pattern with negative gradient. In a quadratic table, the characteristic pattern is symmetry around the turning point."}]},{"id":"b9c2a9a1-5148-463b-9d3c-be53487354e6","cardType":"basic","points":1,"front":"In the coordinate $(3, -2)$, what does the first number represent?","back":"The x-coordinate (movement 3 units right).","mcqOptions":[{"text":"The x-coordinate (movement 3 units right).","correct":true,"explanation":"Coordinates are written in the order $(x,y)$. Therefore, 3 is the x-coordinate and indicates movement 3 units to the right from the origin."},{"text":"The y-coordinate (movement 3 units up).","correct":false,"explanation":"The y-coordinate is the second number, $-2$, and represents vertical movement. The first number, 3, is the x-coordinate and means 3 units right."},{"text":"The gradient (movement 3 units right).","correct":false,"explanation":"A gradient describes the steepness of a line and cannot be read as the first entry of one coordinate. The first number is the x-coordinate."},{"text":"The y-intercept (movement 3 units up).","correct":false,"explanation":"A y-intercept is where a graph crosses the y-axis, not the first number in any coordinate pair. Here, 3 is the x-coordinate."}]},{"id":"475e910a-de11-46cf-a436-dd8c9d9c6d77","cardType":"basic","points":1,"front":"How do you find the y-intercept of a quadratic graph from its equation?","back":"Substitute $x = 0$ into the equation.","mcqOptions":[{"text":"Substitute $x = 0$ into the equation.","correct":true,"explanation":"Every point on the y-axis has an x-coordinate of 0. Substituting $x=0$ therefore calculates the y-value where the quadratic crosses the y-axis."},{"text":"Substitute $y = 0$ into the equation.","correct":false,"explanation":"Setting $y=0$ finds the x-intercepts or roots, where the graph crosses the x-axis. The y-intercept is found by substituting $x=0$."},{"text":"Substitute $x = 1$ into the equation.","correct":false,"explanation":"Substituting $x=1$ finds the point on the graph with x-coordinate 1, not its y-intercept. Points on the y-axis have $x=0$."},{"text":"Substitute $y = 1$ into the equation.","correct":false,"explanation":"Substituting $y=1$ finds possible points at height 1 and does not identify where the graph crosses the y-axis. For that, substitute $x=0$."}]},{"id":"d383fa52-6f2b-4191-acb9-7ef9a615c06b","cardType":"basic","points":1,"front":"What is the value of $y$ at any x-intercept?","back":"$y = 0$","mcqOptions":[{"text":"$y = 0$","correct":true,"explanation":"Every point on the x-axis has a y-coordinate of zero. Therefore, wherever a graph crosses or touches the x-axis, $y = 0$."},{"text":"$x = 0$","correct":false,"explanation":"This confuses the two axes. The condition $x = 0$ identifies points on the y-axis, whereas an x-intercept always has $y = 0$."},{"text":"$y = 1$","correct":false,"explanation":"A y-coordinate of 1 places the point one unit above the x-axis. At an x-intercept, the point lies on the axis, so $y = 0$."},{"text":"$y = -1$","correct":false,"explanation":"A y-coordinate of -1 places the point one unit below the x-axis. An x-intercept lies directly on the axis, where $y = 0$."}]},{"id":"d902631b-e0f7-475d-826e-bd19708e5525","cardType":"basic","points":1,"front":"How does the shape of $y = -x^2 + 4$ differ from $y = x^2 + 4$?","back":"It is an upside-down U-shape (opens downwards).","mcqOptions":[{"text":"It is an upside-down U-shape (opens downwards).","correct":true,"explanation":"The coefficient of $x^2$ is negative, so the parabola opens downwards. The positive version has the same vertex but opens upwards."},{"text":"It is a narrower U-shape (opens upwards).","correct":false,"explanation":"Width depends on the magnitude of the $x^2$ coefficient, which remains 1. The negative sign instead makes the graph an upside-down U-shape opening downwards."},{"text":"It is a wider U-shape (opens upwards).","correct":false,"explanation":"Changing the coefficient from 1 to -1 does not alter its magnitude, so the width is unchanged. The negative sign makes the parabola open downwards."},{"text":"It is a right-shifted U-shape (opens upwards).","correct":false,"explanation":"A horizontal shift would require changing the expression inside a squared bracket. Here, the negative $x^2$ coefficient produces an upside-down U-shape opening downwards."}]},{"id":"5b19674d-8693-4019-b4fb-b32950c315bb","cardType":"cloze","points":null,"front":"The [ ] is the specific point where the parabola changes direction.","back":"The turning point is the specific point where the parabola changes direction.","mcqOptions":[{"text":"turning point","correct":true,"explanation":"The turning point is the point at which a parabola changes from decreasing to increasing, or from increasing to decreasing."},{"text":"axis of symmetry","correct":false,"explanation":"The axis of symmetry is a vertical line dividing the parabola into matching halves, not the specific point where it changes direction. That point is the turning point."},{"text":"x-intercept","correct":false,"explanation":"An x-intercept is where the parabola crosses or touches the x-axis, and it need not change direction there. The direction changes at the turning point."},{"text":"y-intercept","correct":false,"explanation":"The y-intercept is where the graph crosses the y-axis, not necessarily where its direction changes. The specific point of direction change is the turning point."}]},{"id":"05f51136-aacb-419e-aef5-45317e5db685","cardType":"basic","points":1,"front":"Where should the turning point be positioned relative to matching y-values on either side?","back":"Halfway between the symmetric points.","mcqOptions":[{"text":"Halfway between the symmetric points.","correct":true,"explanation":"A parabola is symmetric about the vertical line through its turning point. Two points with equal y-values therefore lie equally far from the turning point."},{"text":"Directly above the left symmetric point.","correct":false,"explanation":"The turning point is not vertically aligned with either matching point. Symmetry places it horizontally halfway between points that have the same y-value."},{"text":"At the same position as one symmetric point.","correct":false,"explanation":"Matching y-values occur at two distinct points on opposite sides of the axis of symmetry. The turning point lies halfway between them, not at either point."},{"text":"Closer to the point with smaller x-value.","correct":false,"explanation":"This would make the two sides asymmetric. Equal y-values occur at equal horizontal distances from the turning point, so it must be halfway between them."}]},{"id":"c8483eaa-b75e-4e65-8275-1999ea6c86e0","cardType":"basic","points":2,"front":"Calculate the y-value for $y = x^2 + 3x - 4$ when $x = -1$.","back":"

$(-1)^2 + 3(-1) - 4$
$1 - 3 - 4 = -6$

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

$(-1)^2 + 3(-1) - 4$
$1 - 3 - 4 = -6$

","correct":true,"explanation":"Substituting $x=-1$ gives $(-1)^2=1$ and $3(-1)=-3$. Combining the terms gives $1-3-4=-6$."},{"text":"

$(-1)^2 + 3(-1) - 4$
$-1 - 3 - 4 = -8$

","correct":false,"explanation":"This treats $(-1)^2$ as -1, but squaring a negative number gives a positive result. Using $(-1)^2=1$ gives $1-3-4=-6$."},{"text":"

$(-1)^2 + 3(-1) - 4$
$1 + 3 - 4 = 0$

","correct":false,"explanation":"This loses the negative sign when calculating $3(-1)$. That product is -3, so the correct calculation is $1-3-4=-6$."},{"text":"

$(-1)^2 + 3(-1) - 4$
$1 - 3 + 4 = 2$

","correct":false,"explanation":"This changes the constant term from -4 to +4. Keeping its original sign gives the correct calculation $1-3-4=-6$."}]},{"id":"be88b4c8-c95d-4144-9536-410d962c214f","cardType":"cloze","points":null,"front":"An [ ] is where the graph crosses the horizontal axis, and a y-intercept is where it crosses the vertical axis.","back":"An x-intercept is where the graph crosses the horizontal axis, and a y-intercept is where it crosses the vertical axis.","mcqOptions":[{"text":"x-intercept","correct":true,"explanation":"The horizontal axis is the x-axis, so a point where the graph crosses it is called an x-intercept. Its y-coordinate is always zero."},{"text":"y-intercept","correct":false,"explanation":"A y-intercept occurs where a graph crosses the vertical y-axis, not the horizontal axis. A crossing of the horizontal axis is an x-intercept."},{"text":"turning point","correct":false,"explanation":"A turning point is where a graph changes direction and may be away from either axis. A crossing of the horizontal axis is an x-intercept."},{"text":"axis of symmetry","correct":false,"explanation":"The axis of symmetry is a line dividing a parabola into matching halves, not a crossing point. The graph crosses the horizontal axis at an x-intercept."}]},{"id":"680363dd-e0c7-4b16-95ba-8a56eb04b6cb","cardType":"cloze","points":null,"front":"An x-intercept is where the graph crosses the horizontal axis, and a [ ] is where it crosses the vertical axis.","back":"An x-intercept is where the graph crosses the horizontal axis, and a y-intercept is where it crosses the vertical axis.","mcqOptions":[{"text":"y-intercept","correct":true,"explanation":"The vertical axis is the y-axis, so the point where a graph crosses it is the y-intercept. Its x-coordinate is always zero."},{"text":"x-intercept","correct":false,"explanation":"An x-intercept is where a graph crosses the horizontal x-axis. The point where it crosses the vertical axis is instead called the y-intercept."},{"text":"turning point","correct":false,"explanation":"A turning point is where the graph changes direction and does not have to lie on the vertical axis. That axis is crossed at the y-intercept."},{"text":"axis of symmetry","correct":false,"explanation":"The axis of symmetry is the line through the centre of a parabola, not its crossing with the vertical axis. That crossing is the y-intercept."}]},{"id":"1964e6ec-6200-4e5c-8b61-ab65ae93dbb8","cardType":"basic","points":1,"front":"If a plotted point does not fit the smooth U-shape of the others, what is the first check?","back":"Recheck the substitution for that specific x-value.","mcqOptions":[{"text":"Recheck the substitution for that specific x-value.","correct":true,"explanation":"An isolated point that breaks the smooth pattern usually indicates a substitution or arithmetic error. Recalculating y using that point’s x-value is the most direct first check."},{"text":"Recheck the scale on the vertical axis.","correct":false,"explanation":"An incorrect axis scale would usually affect several plotted points rather than one isolated point. First recalculate the substitution for the specific x-value that looks wrong."},{"text":"Recheck the curve through every plotted point.","correct":false,"explanation":"A smooth curve should not be forced through an isolated incorrect point. First recheck the substitution for that point’s x-value, then redraw the curve if necessary."},{"text":"Recheck the substitution for a different x-value.","correct":false,"explanation":"Checking a different value does not test the suspicious point. Recalculate the substitution using the specific x-value whose plotted point does not fit the parabola."}]},{"id":"ae168c3e-2f4b-4861-9a55-01c7da3148d9","cardType":"basic","points":1,"front":"What is the common mistake when plotting a coordinate like $(3, -2)$?","back":"Swapping the coordinates (moving vertically before horizontally).","mcqOptions":[{"text":"Swapping the coordinates (moving vertically before horizontally).","correct":true,"explanation":"Coordinates are conventionally plotted by using the first value for horizontal movement and the second for vertical movement. Swapping them places the point at $(-2, 3)$ instead."},{"text":"Keeping the coordinates in order (moving horizontally before vertically).","correct":false,"explanation":"This is the correct plotting order, not the common mistake. The mistake is swapping the coordinates, which would place $(3, -2)$ at $(-2, 3)$."},{"text":"Reading the coordinates in order (using $3$ as $x$ and $-2$ as $y$).","correct":false,"explanation":"Using $3$ as the horizontal coordinate and $-2$ as the vertical coordinate is correct. The common mistake is swapping these coordinate values."},{"text":"Plotting the coordinates accurately (moving right $3$ and down $2$).","correct":false,"explanation":"Moving right $3$ and down $2$ correctly locates $(3, -2)$. The common mistake is swapping the coordinates and plotting $(-2, 3)$ instead."}]},{"id":"ba2a264a-1f96-46cc-8a78-6b3bd920373a","cardType":"cloze","points":null,"front":"For the equation $y = -x^2 + 4$, if $x = -2$, then $y = \\text{[...]}$.","back":"For the equation $y = -x^2 + 4$, if $x = -2$, then $y = \\boxed{0}$.","mcqOptions":[{"text":"0","correct":true,"explanation":"Substituting $x=-2$ gives $y=-(-2)^2+4=-4+4=0$. The square is evaluated before applying the negative sign outside it."},{"text":"8","correct":false,"explanation":"This results from treating $-x^2$ as though the negative value of $x$ makes the term positive. In fact, $-(-2)^2=-4$, so $y=0$."},{"text":"-4","correct":false,"explanation":"This omits the $+4$ after correctly finding $-(-2)^2=-4$. Including the constant gives $y=-4+4=0$."},{"text":"2","correct":false,"explanation":"This comes from failing to square $-2$ correctly, effectively using $x^2=2$. Since $(-2)^2=4$, the calculation is $-4+4=0$."}]},{"id":"aaba646a-5e0b-496b-ab9d-df3841db0cd4","cardType":"basic","points":1,"front":"How far should you draw the curve relative to the plotted points?","back":"Slightly beyond the end points (if the grid allows).","mcqOptions":[{"text":"Slightly beyond the end points (if the grid allows).","correct":true,"explanation":"Extending the smooth curve slightly beyond the first and last plotted points shows that the quadratic continues, while keeping the graph within the available grid."},{"text":"Exactly up to the end points (and no further).","correct":false,"explanation":"Stopping exactly at the end points can wrongly suggest that the quadratic ends there. The curve should extend slightly beyond them if the grid allows."},{"text":"Far beyond the end points (until reaching the grid edges).","correct":false,"explanation":"Extending to the grid edges without supporting plotted values may make the curve inaccurate. It should continue only slightly beyond the end points where possible."},{"text":"Only between the middle points (excluding both end points).","correct":false,"explanation":"Leaving out the end points produces an incomplete graph and ignores known coordinates. The curve should pass through every plotted point and extend slightly beyond the end points."}]},{"id":"f23c7093-edf9-4e92-a79e-9f62de5a3f57","cardType":"cloze","points":null,"front":"A quadratic graph must never be drawn with [ ].","back":"A quadratic graph must never be drawn with straight-line segments.","mcqOptions":[{"text":"straight-line segments","correct":true,"explanation":"A quadratic graph changes gradient continuously, so its plotted points must be joined with a single smooth curve rather than separate straight-line segments."},{"text":"a smooth continuous curve","correct":false,"explanation":"A smooth continuous curve is exactly how a quadratic graph should be drawn. It must never use straight-line segments because its gradient changes continuously."},{"text":"a clearly marked turning point","correct":false,"explanation":"Showing the turning point accurately is appropriate and helps identify the quadratic’s maximum or minimum. Straight-line segments, not a marked turning point, must be avoided."},{"text":"accurately plotted coordinates","correct":false,"explanation":"Accurately plotted coordinates are essential for constructing the graph. The feature that must be avoided is straight-line segments, because a quadratic should form a smooth curve."}]}]}],"examCountdown":null,"heading":"Drawing Quadratic Graphs","title":"Drawing Quadratic Graphs Flashcards","topbar":{"backHref":"/igcse/maths/cie/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"21b9729b-aaab-43f8-bd84-64a0164d1c81","walled":"$L39"}],"$L3a","$L3b"]