Chapter 6 of 8

All 8 chapters
  1. 1Expanding two brackets, readunderstood
  2. 2Factorising quadratics, readunderstood
  3. 3Using factorising, readunderstood
  4. 4Completing the square, readunderstood
  5. 5The quadratic formula, readunderstood
  6. 6Graphs of quadratics, readunderstood
  7. 7Quadratic sequences, readunderstood
  8. 8A line and a curve, readunderstood

M4.9.6 Chapter 6

Graphs of quadratics

I plot the points of y = x² − 2x − 3 on a grid. What shape do they make?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Draw y = x² − 2x − 3 from a table of values
  • Find the roots, turning point and y axis crossing from a graph

A rule like y = x² − 2x − 3 gives a y for every x. To draw its graph, make a table of values. Work out y for several values of x, then plot each pair as a point.

A table with the heads x and y. Rows: −2 and 5; −1 and 0; 0 and −3; 1 and −4; 2 and −3; 3 and 0; 4 and 5.xy−25−100−31−42−33045
The table for y = x² − 2x − 3. Look at how the y values fall, then rise again.

Be careful with negative x. When x = −2, x² is (−2) × (−2) = 4, which is positive. And −2x is −2 × (−2) = +4. So y = 4 + 4 − 3 = 5.

Draw it first

From the table to the curve

  1. A graph with the x axis x and the y axis y. Step 1 plots the seven points of the table as dots. Step 2 joins them with one smooth curve named y = x² − 2x − 3.−2−11234−5−4−3123450xy

    1 Plot each pair from the table as a point. Go across first, then up or down.

  2. A graph with the x axis x and the y axis y. Step 1 plots the seven points of the table as dots. Step 2 joins them with one smooth curve named y = x² − 2x − 3.y = x² − 2x − 3−2−11234−5−4−3123450xy

    2 Join the points with one smooth curve. Never use a ruler, because the graph bends between the points.

The curve of a quadratic is called a parabola. It has a lowest point, or a highest point if the x² term is negative. The curve is a mirror image on the two sides of that point.

Worked example

Make a table for y = x² − 2x − 3 and find the lowest point of the curve.

  1. Work out y when x = −2: (−2) × (−2) − 2 × (−2) − 3 = 4 + 4 − 3 = 5.
  2. Work out y when x = 1: 1 × 1 − 2 × 1 − 3 = 1 − 2 − 3 = −4.
  3. The other values are in the table: y = 0 when x = −1 and x = 3, and y = 5 when x = 4.
  4. The table is symmetric. Look: y = −3 at x = 0 and at x = 2. Also y = 0 at x = −1 and at x = 3. So the turning point is halfway, at x = 1, where y = −4.
  5. So the lowest point is (1, −4).

Answer: The lowest point is (1, −4).

Now read the graph. Look at where the curve meets the axes, and at its lowest point. Each of these tells you something about the quadratic.

See it three ways

With things

Find the points of the curve y = x² − 2x − 3 where it meets the x axis, meets the y axis, and turns.

As a picture

A graph with the x axis x and the y axis y. A curve for y = x² − 2x − 3 crosses the x axis at A and B, crosses the y axis at C, and turns at its lowest point T. A dashed line named x = 1 runs up through T.ABCTx = 1−3−2−1234−5−4−3123450xy

In words

A and B are the roots, where y = 0: x = −1 and x = 3. C is where the curve crosses the y axis: (0, −3). T is the turning point (1, −4), halfway between the roots. The dashed line x = 1 is the line of symmetry.

In symbols

x² − 2x − 3 = (x + 1)(x − 3), so y = 0 at x = −1 and x = 3. At x = 0, y = −3. And x² − 2x − 3 = (x − 1)² − 4, so the lowest point is (1, −4).

So each part of the graph links to work you have done. The roots come from factorising. The y axis crossing is the last number, c. The turning point comes from completing the square.

The turning point is halfway between the roots, because of the mirror symmetry. Halfway between x = −1 and x = 3 is (−1 + 3)/2 = 1. This is a quick way to find x for the turning point. The lowest table value is not always the turning point. Find x by symmetry first.

If the x² term is negative, the curve is upside down. Take y = −x² + 2x + 3. It has a highest point, or maximum, instead of a lowest point.

A graph with the x axis x and the y axis y. The curve for y = −x² + 2x + 3 is shaped like an upside down U. It crosses the x axis at A and B, at −1 and 3, and has its highest point T at (1, 4).ABT−3−2−112345−5−4−3−2−11450xy
The curve of y = −x² + 2x + 3. With −x², it opens downwards. T is a highest point, at (1, 4).

Try it

Draw your own parabola

You need: squared paper, or paper and a ruler, a pencil.

  1. Make a table for y = x² − 4, with x from −3 to 3.
  2. Plot the seven points. Join them with a smooth curve.
  3. Look at the table and the curve. Where are the roots, and where is the lowest point?

What you will see: The y values read the same going up and down the table from the middle. The roots are x = −2 and x = 2. The lowest point is (0, −4).

If you cannot try it, this is what happens.
A table with the heads x and y. Rows: −3 and 5; −2 and 0; −1 and −3; 0 and −4; 1 and −3; 2 and 0; 3 and 5.xy−35−20−1−30−41−32035

A wrong idea many people have

Join the points of a quadratic with a ruler.

Why it feels right: A ruler gives neat lines, and the points are exact, so it seems that joining them is safe.

The test that breaks it: Take x = 0.5. The curve has y = 0.25 − 1 − 3 = −3.75. A ruler line from (0, −3) to (1, −4) gives −3.5 there. They differ.

What is true: A parabola bends between the points. Join them with one smooth curve, and add more points where it turns.

A wrong idea many people have

The lowest point of the curve is where it crosses the y axis.

Why it feels right: The y axis crossing is the one point that is quick to find, so it seems to be the important one.

The test that breaks it: The curve y = x² − 2x − 3 crosses the y axis at (0, −3). But the point (1, −4) is lower.

What is true: The crossing of the y axis is just the point where x = 0. The lowest point is where the curve turns, halfway between the roots.

Key words

parabola
the smooth curve of a quadratic, shaped like a U or an upside down U.
turning point
the lowest point of a U shaped curve, or the highest point of an upside down one.
line of symmetry
a line that cuts a shape into two mirror images, such as x = 1 for this curve.

Back to your guess

The answer is B: A smooth curve with a lowest point, shaped like a U.

The x² makes the graph bend. Its points lie on a smooth curve, shaped like a U, called a parabola. The curve is lowest at the point (1, −4), and there it turns, with no sharp point.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Find y when x = −3 for y = x² − 2x − 3.

    Tap your answer.

    1. Not this one. The square (−3)² is 9, not −9. A negative times a negative is positive. Go to M4.9.6. Try another.

    2. Yes, that is right. x² = (−3) × (−3) = 9, and −2x = −2 × (−3) = 6. So y = 9 + 6 − 3 = 12.

    3. Not this one. The term −2x is −2 × (−3) = +6, not −6. Go to M4.9.6. Try another.

    4. Not this one. Both signs slipped. (−3)² is +9 and −2x is +6, so y = 9 + 6 − 3 = 12. Go to M4.9.6. Try another.

    See the answer

    12

    x² = (−3) × (−3) = 9, and −2x = −2 × (−3) = 6. So y = 9 + 6 − 3 = 12.

    • If you chose A: The square (−3)² is 9, not −9. A negative times a negative is positive. Go to M4.9.6.
    • If you chose C: The term −2x is −2 × (−3) = +6, not −6. Go to M4.9.6.
    • If you chose D: Both signs slipped. (−3)² is +9 and −2x is +6, so y = 9 + 6 − 3 = 12. Go to M4.9.6.

  2. The graph shows a quadratic curve. Where does it cross the x axis?

    A graph with the x axis x and the y axis y. A curve named y = x² − 4x + 3 dips below the x axis between two crossing points.y = x² − 4x + 3134−2−1123450xy
    Tap your answer.

    1. Yes, that is right. The curve meets the x axis where y = 0. From the picture, that is at x = 1 and at x = 3.

    2. Not this one. The curve crosses the axis on the right of the y axis, where x is positive. Go to M4.9.6. Try another.

    3. Not this one. The point (0, 3) is where the curve crosses the y axis. The roots are values of x where y = 0. Go to M4.9.6. Try another.

    4. Not this one. The value x = 2 is the line of symmetry, halfway between the roots. The curve is below the x axis there. Go to M4.9.6. Try another.

    See the answer

    x = 1 and x = 3

    The curve meets the x axis where y = 0. From the picture, that is at x = 1 and at x = 3.

    • If you chose B: The curve crosses the axis on the right of the y axis, where x is positive. Go to M4.9.6.
    • If you chose C: The point (0, 3) is where the curve crosses the y axis. The roots are values of x where y = 0. Go to M4.9.6.
    • If you chose D: The value x = 2 is the line of symmetry, halfway between the roots. The curve is below the x axis there. Go to M4.9.6.

  3. The curve y = (x − 2)² − 1 has a lowest point. What are its coordinates?

    Tap your answer.

    1. Not this one. The square (x − 2)² is 0 when x = 2, not −2. Take care with the sign inside the bracket. Go to M4.9.4. Try another.

    2. Not this one. The number after the square is −1, so the lowest height is −1, not 1. Go to M4.9.4. Try another.

    3. Yes, that is right. A square is never below 0, and (x − 2)² is 0 when x = 2. Then y = 0 − 1 = −1. So the lowest point is (2, −1).

    4. Not this one. You have swapped the two numbers. A point is written (x, y): across first, then up or down. Go to M4.9.6. Try another.

    See the answer

    (2, −1)

    A square is never below 0, and (x − 2)² is 0 when x = 2. Then y = 0 − 1 = −1. So the lowest point is (2, −1).

    • If you chose A: The square (x − 2)² is 0 when x = 2, not −2. Take care with the sign inside the bracket. Go to M4.9.4.
    • If you chose B: The number after the square is −1, so the lowest height is −1, not 1. Go to M4.9.4.
    • If you chose D: You have swapped the two numbers. A point is written (x, y): across first, then up or down. Go to M4.9.6.

  4. Where does the curve y = x² + 3x − 10 cross the y axis?

    Tap your answer.

    1. Not this one. You have swapped x and y. On the y axis, x is 0, so the point is (0, something). Go to M4.9.6. Try another.

    2. Not this one. This point is on the x axis, where y = 0. It is a root of the curve, not its crossing of the y axis. Go to M4.9.3. Try another.

    3. Not this one. The number 3 is the number in front of x. At x = 0, the curve has y = 0 + 0 − 10. Go to M4.9.6. Try another.

    4. Yes, that is right. On the y axis x = 0. Then y = 0 + 0 − 10 = −10. The crossing is (0, −10), and −10 is the last number of the quadratic.

    See the answer

    (0, −10)

    On the y axis x = 0. Then y = 0 + 0 − 10 = −10. The crossing is (0, −10), and −10 is the last number of the quadratic.

    • If you chose A: You have swapped x and y. On the y axis, x is 0, so the point is (0, something). Go to M4.9.6.
    • If you chose B: This point is on the x axis, where y = 0. It is a root of the curve, not its crossing of the y axis. Go to M4.9.3.
    • If you chose C: The number 3 is the number in front of x. At x = 0, the curve has y = 0 + 0 − 10. Go to M4.9.6.

  5. What is the shape of the curve y = 4 − x²?

    Tap your answer.

    1. Not this one. The x² term is −x², which is negative. That turns the U upside down, so the curve has a highest point. Go to M4.9.6. Try another.

    2. Yes, that is right. The x² term is negative, so the curve opens downwards. Its highest point is (0, 4), and it crosses the x axis at −2 and 2.

    3. Not this one. A straight line comes from a rule like y = mx + c. A rule with x² gives a curve. Go to M4.9.6. Try another.

    See the answer

    An upside down U, with a highest point.

    The x² term is negative, so the curve opens downwards. Its highest point is (0, 4), and it crosses the x axis at −2 and 2.

    • If you chose A: The x² term is −x², which is negative. That turns the U upside down, so the curve has a highest point. Go to M4.9.6.
    • If you chose C: A straight line comes from a rule like y = mx + c. A rule with x² gives a curve. Go to M4.9.6.

  6. Amina plots the seven points of y = x² − 2x − 3 and joins them with a ruler. What is wrong?

    Tap your answer.

    1. Not this one. The points are exact, but between them the true graph bends. A ruler line has the wrong shape there. Go to M4.9.6. Try another.

    2. Not this one. More points help, but between any two points the graph is curved. A ruler is still wrong. Join the points with a smooth curve. Go to M4.9.6. Try another.

    3. Yes, that is right. A quadratic graph bends between the plotted points. Straight pieces cut across the bend, so they are not on the curve.

    See the answer

    The graph is a smooth curve. Straight pieces cut the corners and miss the true curve.

    A quadratic graph bends between the plotted points. Straight pieces cut across the bend, so they are not on the curve.

    • If you chose A: The points are exact, but between them the true graph bends. A ruler line has the wrong shape there. Go to M4.9.6.
    • If you chose B: More points help, but between any two points the graph is curved. A ruler is still wrong. Join the points with a smooth curve. Go to M4.9.6.

  7. A parabola has the line of symmetry x = 3. It crosses the x axis at x = 8. Where else does it cross?

    Tap your answer.

    1. Yes, that is right. The root x = 8 is 5 to the right of the line x = 3. Its mirror image is 5 to the left of 3, which is x = −2.

    2. Not this one. That mirrors 8 in the y axis, not in the line x = 3. The mirror image is as far from x = 3 as 8 is. Go to M4.9.6. Try another.

    3. Not this one. The number 14 is on the same side of the line as 8. The mirror image is on the other side. Go to M4.9.6. Try another.

    See the answer

    x = −2

    The root x = 8 is 5 to the right of the line x = 3. Its mirror image is 5 to the left of 3, which is x = −2.

    • If you chose B: That mirrors 8 in the y axis, not in the line x = 3. The mirror image is as far from x = 3 as 8 is. Go to M4.9.6.
    • If you chose C: The number 14 is on the same side of the line as 8. The mirror image is on the other side. Go to M4.9.6.

  8. From an earlier chapter

    Solve x² − 8x + 12 = 0.

    Tap your answer.

    1. Not this one. The brackets are (x − 2)(x − 6). Putting x − 2 = 0 gives x = 2, not −2. Go to M4.9.3. Try another.

    2. Not this one. These are the numbers in the equation. The solutions come from the brackets, and they are 2 and 6. Go to M4.9.2. Try another.

    3. Not this one. Both numbers in the brackets are negative, (x − 2)(x − 6), so both solutions are positive. Go to M4.9.3. Try another.

    4. Yes, that is right. Two numbers with product 12 and sum −8 are −2 and −6. So (x − 2)(x − 6) = 0, and x = 2 or x = 6.

    See the answer

    x = 2 or x = 6

    Two numbers with product 12 and sum −8 are −2 and −6. So (x − 2)(x − 6) = 0, and x = 2 or x = 6.

    • If you chose A: The brackets are (x − 2)(x − 6). Putting x − 2 = 0 gives x = 2, not −2. Go to M4.9.3.
    • If you chose B: These are the numbers in the equation. The solutions come from the brackets, and they are 2 and 6. Go to M4.9.2.
    • If you chose C: Both numbers in the brackets are negative, (x − 2)(x − 6), so both solutions are positive. Go to M4.9.3.

  9. Find y when x = 3 for y = x² − 4x + 5.

    See the answer

    2

    x² = 9 and −4x = −12. So y = 9 − 12 + 5 = 2.

  10. A parabola crosses the x axis at x = 2 and x = 8. At which value of x is its turning point?

    See the answer

    5

    The turning point is halfway between the roots. Halfway between 2 and 8 is (2 + 8)/2 = 5.

Keep it

Copy this into your exercise book.

Make a table of values, plot the points, join them with a smooth curve. Roots: where y = 0. Crossing of the y axis: the last number. Turning point: halfway between the roots.

Draw: A U shaped curve with its two roots, its y axis crossing, and its turning point marked and named.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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