Chapter 6 of 8

All 8 chapters
  1. 1Function notation, readunderstood
  2. 2Composite and inverse functions, readunderstood
  3. 3Cubic, reciprocal and circle graphs, readunderstood
  4. 4Exponential graphs, readunderstood
  5. 5Graphs of sin, cos and tan, readunderstood
  6. 6Moving graphs, readunderstood
  7. 7Gradients and areas under graphs, readunderstood
  8. 8Solving with graphs and trials, readunderstood

M4.10.6 Chapter 6

Moving graphs

If I draw y = f(x + 3), does the graph of y = f(x) slide to the right or to the left?

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

  • Sketch y = f(x) + a and y = f(x + a)
  • Sketch y = −f(x) and y = f(−x)

Once you know the shape of a graph, you can move it about. We start with one curve, y = f(x), where f(x) = x² − 2x. Its lowest point is V at (1, −1), because f(1) = 1 − 2 = −1.

Follow V as the curve moves. Each change to the rule moves V in a different way. The shape of the curve stays the same, unless the curve is turned over.

First, add a number after the function. For y = f(x) + 3, every output is 3 more than before. So every point of the curve goes up by 3. A slide like this is called a translation.

A graph with the x axis x and the y axis y. A solid curve for y = f(x) has its lowest point V at (1, −1). A dashed curve for y = f(x) + 3 is the same shape, with its lowest point W at (1, 2). A dashed upright line joins V and W. A key under the graph names the solid curve y = f(x) and the dashed curve y = f(x) + 3.VW−2−1234−4−2260xyy = f(x)y = f(x) + 3
Adding 3 outside the function moves the whole curve up 3. V goes to W.

Worked example

Find the lowest point of y = f(x) + 3.

  1. The lowest point of y = f(x) is V at (1, −1), because f(1) = −1.
  2. For y = f(x) + 3, at x = 1 the output is f(1) + 3 = −1 + 3.
  3. So −1 + 3 = 2, and the lowest point moves up to (1, 2).

Answer: The lowest point is (1, 2).

Now add a number inside the brackets. For y = f(x + 3), the input is changed before f works on it. The result may be a surprise: the curve slides to the left.

Here is why. V is the lowest point because f works on the input 1. For y = f(x + 3) we need x + 3 = 1, which gives x = −2. So the lowest point now stands at x = −2. The graph has moved 3 to the left.

A graph with the x axis x and the y axis y. A solid curve for y = f(x) has its lowest point V at (1, −1). A dashed curve for y = f(x + 3) is the same shape, with its lowest point W at (−2, −1), 3 units to the left. A dashed flat line joins W and V. A key under the graph names the solid curve y = f(x) and the dashed curve y = f(x + 3).VW−5−4234−4−2460xyy = f(x)y = f(x + 3)
Adding 3 inside the brackets moves the curve 3 to the left. V goes to W.

Worked example

Find the lowest point of y = f(x + 3).

  1. The lowest point of y = f(x) is where the input of f is 1.
  2. For y = f(x + 3), we need x + 3 = 1. So x = 1 − 3 = −2.
  3. At x = −2 the output is f(−2 + 3) = f(1) = −1.
  4. So the lowest point moves to (−2, −1), which is 3 to the left of V.

Answer: The lowest point is (−2, −1).

The plus sign inside the brackets moves the graph left, and a minus sign moves it right. So y = f(x − 2) slides the graph 2 to the right.

Next, the flips. The rule y = −f(x) changes the sign of every output. A point (x, y) becomes (x, −y). So the curve is turned over, like a mirror image in the x axis. We call this a reflection. Points on the x axis stay where they are.

A graph with the x axis x and the y axis y. A solid curve for y = f(x) is a U with its lowest point V at (1, −1). A dashed curve for y = −f(x) is an upside down U with its highest point W at (1, 1). Both curves cross the x axis at 0 and 2. A key under the graph names the solid curve y = f(x) and the dashed curve y = −f(x).VW−123−4−224xyy = f(x)y = −f(x)
The curve y = −f(x) is y = f(x) turned over in the x axis. V goes to W.

The rule y = f(−x) changes the sign of every input. A point (x, y) becomes (−x, y). So the curve is turned over like a reflection in the y axis. Points on the y axis stay where they are.

A graph with the x axis x and the y axis y. A solid curve for y = f(x) has its lowest point V at (1, −1). A dashed curve for y = f(−x) is its mirror image on the other side of the y axis, with its lowest point W at (−1, −1). Both curves pass through the origin. A key under the graph names the solid curve y = f(x) and the dashed curve y = f(−x).VW−3−113−224xyy = f(x)y = f(−x)
The curve y = f(−x) is y = f(x) turned over in the y axis. V goes to W.

Worked example

The curve y = f(x) passes through the point (3, 3). Find the matching points on y = −f(x) and on y = f(−x).

  1. First check that (3, 3) is on the curve. f(3) = 3² − 2 × 3 = 9 − 6 = 3.
  2. For y = −f(x) keep x = 3 and change the sign of y. So −f(3) = −3.
  3. The matching point on y = −f(x) is (3, −3).
  4. For y = f(−x) change the sign of x. At x = −3, f(−x) = f(3) = 3. The matching point is (−3, 3).

Answer: The points are (3, −3) and (−3, 3).

See it three ways

With things

A path is drawn on a map. The whole map is slid 3 squares up. Every point of the path stays in its column, and is 3 squares higher.

As a picture

A table with the heads x, f(x) and f(x) + 3. Rows: 0 and 0 and 3; 1 and −1 and 2; 2 and 0 and 3; 3 and 3 and 6.xf(x)f(x) + 30031−12203336

In words

Each output has 3 added to it. Each x stays the same. So every point moves straight up by 3.

In symbols

y = f(x) + 3, so (1, −1) moves to (1, 2).

Try it

Slide and flip a curve

You need: squared paper, thin paper that you can see through, a pencil.

  1. Draw y = x² − 2x on squared paper, and mark its lowest point V at (1, −1). Trace the curve and the axes onto the thin paper.
  2. Slide the thin paper 3 up, keeping the axes level, and mark where V lands. Put it back, then slide it 3 to the left and mark V again.
  3. Put it back. Fold the thin paper along the x axis and mark V. Unfold it, then fold along the y axis and mark V.

What you will see: Slides keep the curve the same way up. Folding along the x axis turns it upside down. Folding along the y axis turns it left to right.

If you cannot try it, this is what happens.
A table with the heads move and V. Rows: start and (1, −1); up 3 and (1, 2); left 3 and (−2, −1); fold on x axis and (1, 1); fold on y axis and (−1, −1).moveVstart(1, −1)up 3(1, 2)left 3(−2, −1)fold on x axis(1, 1)fold on y axis(−1, −1)

A wrong idea many people have

y = f(x + 3) moves the graph 3 to the right.

Why it feels right: Adding 3 usually makes things larger or moves them to the right along a number line.

The test that breaks it: The lowest point of y = f(x) is at x = 1. For y = f(x + 3), the lowest point needs x + 3 = 1, so it is at x = −2, which is left of 1.

What is true: A number added inside the brackets moves the graph the opposite way. The graph of f(x + 3) moves left 3. The graph of f(x − 3) moves right 3.

A wrong idea many people have

y = −f(x) and y = f(−x) are the same.

Why it feels right: Both have a minus sign next to f, and both seem to turn the graph round.

The test that breaks it: For f(x) = x² − 2x, the point V is at (1, −1). On y = −f(x) it goes to (1, 1). On y = f(−x) it goes to (−1, −1). They are different points.

What is true: A minus outside, as in −f(x), turns the graph over in the x axis. A minus inside, as in f(−x), turns it over in the y axis.

Key words

translation
a slide of a graph without turning it, such as 3 up or 3 left.
reflection
a turn over of a graph, as in a mirror, such as in the x axis or the y axis.

Back to your guess

The answer is B: To the left, even though 3 is added to x.

The graph slides 3 to the left. The curve y = f(x + 3) gives at x = −2 the same output that y = f(x) gives at x = 1. So every point of the curve arrives 3 earlier on the x axis, which is to the left.

Check yourself

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

  1. The curve y = f(x) has its lowest point at (2, 5). Where is the lowest point of y = f(x) − 4?

    Tap your answer.

    1. Yes, that is right. The x value stays at 2. Every output has 4 taken away, so y goes from 5 to 5 − 4 = 1.

    2. Not this one. You moved the point left, but subtracting 4 outside the function changes only the y value, which is 5 − 4. Go to M4.10.6. Try another.

    3. Not this one. You moved the point right. A number outside the function moves the graph up or down, not sideways. Go to M4.10.6. Try another.

    See the answer

    (2, 1)

    The x value stays at 2. Every output has 4 taken away, so y goes from 5 to 5 − 4 = 1.

    • If you chose B: You moved the point left, but subtracting 4 outside the function changes only the y value, which is 5 − 4. Go to M4.10.6.
    • If you chose C: You moved the point right. A number outside the function moves the graph up or down, not sideways. Go to M4.10.6.

  2. The curve y = f(x) has its lowest point at (2, 5). Where is the lowest point of y = f(x + 4)?

    Tap your answer.

    1. Not this one. You moved the point right. A number added inside the brackets moves the graph the opposite way, so left. Go to M4.10.6. Try another.

    2. Not this one. You added 4 to the y value. That is what f(x) + 4 does. With the 4 inside the brackets, the graph moves sideways. Go to M4.10.6. Try another.

    3. Yes, that is right. The lowest point is where the input of f is 2. So we need x + 4 = 2, which gives x = −2. The height stays 5.

    See the answer

    (−2, 5)

    The lowest point is where the input of f is 2. So we need x + 4 = 2, which gives x = −2. The height stays 5.

    • If you chose A: You moved the point right. A number added inside the brackets moves the graph the opposite way, so left. Go to M4.10.6.
    • If you chose B: You added 4 to the y value. That is what f(x) + 4 does. With the 4 inside the brackets, the graph moves sideways. Go to M4.10.6.

  3. The dashed curve is a moved copy of the solid curve y = f(x). Which equation gives the dashed curve?

    A graph with the x axis x and the y axis y. A solid U shaped curve for y = f(x) has its lowest point at (1, −1). A dashed curve of the same shape has its lowest point at (5, −1), four units to the right. A key under the graph names the solid curve y = f(x) and the dashed curve the curve to find.−2246−2460xyy = f(x)the curve to find
    Tap your answer.

    1. Not this one. That would move the curve up by 4. The dashed curve is at the same height as the solid one, so it has moved sideways. Go to M4.10.6. Try another.

    2. Not this one. That would move the curve 4 to the left, because a plus inside the brackets moves the graph left. The dashed curve is on the right. Go to M4.10.6. Try another.

    3. Not this one. That would turn the curve upside down. The dashed curve is the same way up as the solid one. Go to M4.10.6. Try another.

    4. Yes, that is right. The dashed curve is the solid curve slid 4 to the right, with its lowest point at (5, −1). A minus inside the brackets moves the graph right, so it is y = f(x − 4).

    See the answer

    y = f(x − 4)

    The dashed curve is the solid curve slid 4 to the right, with its lowest point at (5, −1). A minus inside the brackets moves the graph right, so it is y = f(x − 4).

    • If you chose A: That would move the curve up by 4. The dashed curve is at the same height as the solid one, so it has moved sideways. Go to M4.10.6.
    • If you chose B: That would move the curve 4 to the left, because a plus inside the brackets moves the graph left. The dashed curve is on the right. Go to M4.10.6.
    • If you chose C: That would turn the curve upside down. The dashed curve is the same way up as the solid one. Go to M4.10.6.

  4. Which rule turns the graph of y = f(x) upside down, in the x axis?

    Tap your answer.

    1. Not this one. That turns the graph left to right, in the y axis. The minus is inside the brackets, so it changes the inputs. Go to M4.10.6. Try another.

    2. Yes, that is right. The rule −f(x) changes the sign of every output. So points above the x axis go below it, and the graph turns upside down.

    3. Not this one. That moves the graph down a little, but it is still the same way up. It does not turn the graph over. Go to M4.10.6. Try another.

    See the answer

    y = −f(x)

    The rule −f(x) changes the sign of every output. So points above the x axis go below it, and the graph turns upside down.

    • If you chose A: That turns the graph left to right, in the y axis. The minus is inside the brackets, so it changes the inputs. Go to M4.10.6.
    • If you chose C: That moves the graph down a little, but it is still the same way up. It does not turn the graph over. Go to M4.10.6.

  5. The curve y = f(x) passes through the point (3, 4). Which point is on the curve y = f(−x)?

    Tap your answer.

    1. Yes, that is right. For y = f(−x), at x = −3 the output is f(3) = 4. So the point (−3, 4) is on the curve. Only the x value changes sign.

    2. Not this one. That is the matching point on y = −f(x), where the sign of y changes. For f(−x) the sign of x changes. Go to M4.10.6. Try another.

    3. Not this one. You changed the sign of both numbers. That is a turn about the origin. For f(−x) only x changes sign. Go to M4.10.6. Try another.

    See the answer

    (−3, 4)

    For y = f(−x), at x = −3 the output is f(3) = 4. So the point (−3, 4) is on the curve. Only the x value changes sign.

    • If you chose B: That is the matching point on y = −f(x), where the sign of y changes. For f(−x) the sign of x changes. Go to M4.10.6.
    • If you chose C: You changed the sign of both numbers. That is a turn about the origin. For f(−x) only x changes sign. Go to M4.10.6.

  6. Which rule moves the graph of y = f(x) up by 5?

    Tap your answer.

    1. Not this one. The 5 is inside the brackets, so it moves the graph sideways, 5 to the right, not up. Go to M4.10.6. Try another.

    2. Yes, that is right. To move every point up by 5, add 5 to every output, outside the function. That is y = f(x) + 5.

    3. Not this one. This multiplies every output by 5, which stretches the graph. It does not slide it up by 5. Go to M4.10.6. Try another.

    See the answer

    y = f(x) + 5

    To move every point up by 5, add 5 to every output, outside the function. That is y = f(x) + 5.

    • If you chose A: The 5 is inside the brackets, so it moves the graph sideways, 5 to the right, not up. Go to M4.10.6.
    • If you chose C: This multiplies every output by 5, which stretches the graph. It does not slide it up by 5. Go to M4.10.6.

  7. From an earlier chapter

    The equation sin x = 0.5 has the solutions 30° and 150°, between 0° and 360°. What are the solutions of sin x = −0.5?

    Tap your answer.

    1. Not this one. These ignore the minus sign. The value −0.5 is below the axis, where the sin wave is for angles between 180° and 360°. Go to M4.10.5. Try another.

    2. Not this one. The angle 150° has sin 150° = 0.5, which is positive. Both angles must be between 180° and 360°. Go to M4.10.5. Try another.

    3. Yes, that is right. The calculator gives sin⁻¹(−0.5) = −30°. On the wave the angles are 180° + 30° = 210° and 360° − 30° = 330°.

    See the answer

    x = 210° or x = 330°

    The calculator gives sin⁻¹(−0.5) = −30°. On the wave the angles are 180° + 30° = 210° and 360° − 30° = 330°.

    • If you chose A: These ignore the minus sign. The value −0.5 is below the axis, where the sin wave is for angles between 180° and 360°. Go to M4.10.5.
    • If you chose B: The angle 150° has sin 150° = 0.5, which is positive. Both angles must be between 180° and 360°. Go to M4.10.5.

  8. The lowest point of y = f(x) is at (4, −3). What is the y value of the lowest point of y = f(x) + 5?

    See the answer

    2

    Adding 5 outside the function moves every point up 5. So the y value goes from −3 to −3 + 5 = 2.

  9. The lowest point of y = f(x) is at (5, 2). What is the x value of the lowest point of y = f(x + 3)?

    See the answer

    2

    We need x + 3 = 5, so x = 5 − 3 = 2. The graph moves 3 to the left.

Keep it

Copy this into your exercise book.

f(x) + a: up a. f(x + a): left a. −f(x): over in the x axis. f(−x): over in the y axis. Follow one point to check.

Draw: A curve and its four moved copies, with one marked point on each.

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

No internet tomorrow? Save the whole book as one file (956 KB). It opens on any phone.