Chapter 8 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

M4.10.8 Chapter 8

Solving with graphs and trials

Two graphs cross at a point. What does that point have to do with solving an equation?

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

  • Solve an equation by finding where two graphs meet
  • Find a root between two numbers by trying values

Some equations are hard to solve with algebra, but a graph can help. Take x² = x + 2. Draw two graphs, y = x² and y = x + 2. At a point where they cross, the two y values are equal. So x² = x + 2 there.

In general, to solve f(x) = g(x), draw y = f(x) and y = g(x) on one grid. The x values of the crossing points are the solutions. Each crossing point is called an intersection.

A graph with the x axis x and the y axis y. A solid U shaped curve for y = x² and a dashed straight line for y = x + 2 cross at two points, A at (−1, 1) and B at (2, 4). A key under the graph names the solid curve y = x² and the dashed line y = x + 2.AB−2−1123−24680xyy = x²y = x + 2
The graphs of y = x² and y = x + 2 cross at A and B.

Worked example

Use the graphs of y = x² and y = x + 2 to solve x² = x + 2.

  1. Draw y = x² and y = x + 2 on one grid.
  2. The graphs cross at A (−1, 1) and B (2, 4).
  3. The x values of the crossing points are the solutions: x = −1 and x = 2.
  4. Check x = 2: x² = 4, and x + 2 = 2 + 2 = 4. The two sides agree.

Answer: x = −1 or x = 2.

The same idea works for any pair of graphs. To solve 2ˣ = 10, draw y = 2ˣ and the flat line y = 10, and read the x value where they meet. A graph gives an estimate, because you can read it only to the accuracy of the drawing.

To get closer, use trials. A root of f is a value of x with f(x) = 0. It is where the graph of y = f(x) crosses the x axis. Take f(x) = x³ + x − 3. Its graph crosses the x axis once, near x = 1.2.

A graph with the x axis x and the y axis y. A curve for y = x³ + x − 3 rises from below the x axis at the left to above it at the right. It crosses the x axis at the point R, near x = 1.2.R1.52−4−224680xy
The graph of y = x³ + x − 3 crosses the x axis at R, so R is a root of f.

Work out f(x) for some values of x. Suppose f is negative at one value and positive at another. If the graph has no break between them, it must cross the x axis between those values. There is a root there.

Worked example

f(x) = x³ + x − 3. Show that f has a root between 1 and 2.

  1. Work out f at x = 1 and x = 2.
  2. f(1) = 1 + 1 − 3 = −1.
  3. f(2) = 8 + 2 − 3 = 7.
  4. f(1) is negative and f(2) is positive. The graph has no break, so it crosses the x axis between 1 and 2.

Answer: There is a root between 1 and 2.

Now close in on the root. Try values between 1 and 2, in steps of 0.1. Look for the place where f changes sign.

Worked example

f(x) = x³ + x − 3 has a root between 1 and 2. Find it to one decimal place.

  1. Try values in steps of 0.1. We look for a change of sign.
  2. f(1.2) = 1.728 + 1.2 − 3 = −0.072, which is negative.
  3. f(1.3) = 2.197 + 1.3 − 3 = 0.497, which is positive.
  4. The root is between 1.2 and 1.3. To decide how to round, try the middle value: f(1.25) = 1.953125 + 1.25 − 3 = 0.203125.
  5. f(1.25) is positive, so the root is between 1.2 and 1.25. It is below 1.25, so to one decimal place the root is 1.2.

Answer: x = 1.2 (to one decimal place).

The middle value matters. Between 1.2 and 1.3, the answer to one decimal place could be 1.2 or 1.3. Trying the middle value, 1.25, shows on which side of it the root lies.

See it three ways

With things

Amina has a plant 10 cm high that grows 2 cm each day. Ravi has a plant 4 cm high that grows 4 cm each day. When are the two plants the same height?

As a picture

A graph with the x axis days and the y axis height (cm). A solid line for Amina starts at 10 and rises slowly. A dashed line for Ravi starts at 4 and rises faster. They cross at the point (3, 16), which has dashed lines to both axes. A key under the graph names the solid line Amina and the dashed line Ravi.123456510152025300daysheight (cm)AminaRavi

In words

The lines cross where the two heights are equal. That is after 3 days, when both plants are 16 cm high.

In symbols

10 + 2d = 4 + 4d, so 6 = 2d and d = 3. The height is 10 + 2 × 3 = 16 cm.

Try it

Close in on a square root

You need: a calculator, or paper and a pencil.

  1. We want x with x² = 7. Try x = 2 and x = 3 first.
  2. Then try 2.5, 2.6, 2.7 and 2.65. For each, write whether x² is lower or higher than 7.
  3. Find the two values closest together that have 7 between their squares.

What you will see: 2.6² = 6.76 is low and 2.65² = 7.0225 is high, so the solution is between 2.6 and 2.65. To one decimal place, it is 2.6.

If you cannot try it, this is what happens.
A table with the heads x, x² and vs 7. Rows: 2 and 4 and low; 3 and 9 and high; 2.5 and 6.25 and low; 2.6 and 6.76 and low; 2.7 and 7.29 and high; 2.65 and 7.0225 and high.xx²vs 724low39high2.56.25low2.66.76low2.77.29high2.657.0225high

A wrong idea many people have

The solutions of x² = x + 2 are the y values of the crossing points.

Why it feels right: The crossing points are where the graphs meet, and the heights seem to be the answers.

The test that breaks it: The crossing points are (−1, 1) and (2, 4). Put x = 4 into the equation: 4² = 16, but 4 + 2 = 6. They are not equal. Put x = 2: 2² = 4 and 2 + 2 = 4.

What is true: The equation asks for the x values where the two rules agree. Read the x value of each crossing point, going down to the x axis.

A wrong idea many people have

If f changes sign between two numbers, there must be a root between them.

Why it feels right: The graph goes from below the axis to above it, so it seems it must cross the axis.

The test that breaks it: Take f(x) = 1/x. Then f(−1) = −1 and f(1) = 1, a change of sign. But 1/x is never zero. The graph has a break at x = 0 and jumps across the axis without touching it.

What is true: A sign change shows a root only when the graph has no break between the two numbers. Graphs made only of numbers and positive whole number powers of x, such as x³ + x − 3, have no breaks.

Key words

intersection
a point where two graphs cross.
root
a value of x for which f(x) = 0. It is where the graph of y = f(x) meets the x axis.

Back to your guess

The answer is C: The x value of the point is a solution of the equation.

At a crossing point, the two graphs have the same x and the same y. So at that x, the two rules give the same output. That means the x value solves the equation that says the two rules are equal.

Check yourself

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

  1. The graphs of y = x² and y = 2x + 3 cross at the points shown. What are the solutions of x² = 2x + 3?

    A graph with the x axis x and the y axis y. A solid U shaped curve for y = x² and a dashed straight line for y = 2x + 3 cross at two points, C at (−1, 1) and D at (3, 9). A key under the graph names the solid curve y = x² and the dashed line y = 2x + 3.CD−11234−2468100xyy = x²y = 2x + 3
    Tap your answer.

    1. Yes, that is right. The solutions are the x values of the crossing points C (−1, 1) and D (3, 9). Check x = 3: x² = 9, and 2x + 3 = 9.

    2. Not this one. These are the y values of the crossing points. The solutions are the x values, which are read across the bottom. Go to M4.10.8. Try another.

    3. Not this one. You took the x value of C and the y value of D. Both solutions must be x values: −1 and 3. Go to M4.10.8. Try another.

    See the answer

    x = −1 or x = 3

    The solutions are the x values of the crossing points C (−1, 1) and D (3, 9). Check x = 3: x² = 9, and 2x + 3 = 9.

    • If you chose B: These are the y values of the crossing points. The solutions are the x values, which are read across the bottom. Go to M4.10.8.
    • If you chose C: You took the x value of C and the y value of D. Both solutions must be x values: −1 and 3. Go to M4.10.8.

  2. f(x) = x² − 5. Then f(2) = −1 and f(3) = 4. Where does f have a root?

    Tap your answer.

    1. Not this one. A value close to 0 is not a root. A root needs f(x) to be exactly 0, and f(2) is −1. Go to M4.10.8. Try another.

    2. Not this one. The numbers −1 and 4 are the outputs, the values of f. The root is a value of x, so look between the inputs 2 and 3. Go to M4.10.8. Try another.

    3. Yes, that is right. f(2) is negative and f(3) is positive, and the graph of x² − 5 has no break. So it crosses the x axis between 2 and 3.

    See the answer

    Between 2 and 3.

    f(2) is negative and f(3) is positive, and the graph of x² − 5 has no break. So it crosses the x axis between 2 and 3.

    • If you chose A: A value close to 0 is not a root. A root needs f(x) to be exactly 0, and f(2) is −1. Go to M4.10.8.
    • If you chose B: The numbers −1 and 4 are the outputs, the values of f. The root is a value of x, so look between the inputs 2 and 3. Go to M4.10.8.

  3. For f(x) = x² − 5 we find f(2) = −1, f(2.5) = 1.25 and f(3) = 4. Where is the root now?

    Tap your answer.

    1. Not this one. We know f(3) is positive and f(2.5) is positive too, so there is no sign change there. The sign changes between 2 and 2.5. Go to M4.10.8. Try another.

    2. Yes, that is right. f(2) is negative and f(2.5) is positive, so the sign changes between 2 and 2.5. The root is in that smaller gap.

    3. Not this one. At 2.5 the value is 1.25, not 0. So 2.5 is not a root. Go to M4.10.8. Try another.

    See the answer

    Between 2 and 2.5

    f(2) is negative and f(2.5) is positive, so the sign changes between 2 and 2.5. The root is in that smaller gap.

    • If you chose A: We know f(3) is positive and f(2.5) is positive too, so there is no sign change there. The sign changes between 2 and 2.5. Go to M4.10.8.
    • If you chose C: At 2.5 the value is 1.25, not 0. So 2.5 is not a root. Go to M4.10.8.

  4. The graphs of y = 2ˣ and y = 20 are drawn. Which is the best estimate of the solution of 2ˣ = 20?

    A graph with the x axis x and the y axis y. A curve for y = 2ˣ rises from the left, steeper and steeper. A dashed flat line at height 20 meets the curve at the point S, with dashed lines to both axes.S1234581624320xy
    Tap your answer.

    1. Not this one. The number 20 is the height of the flat line, the value of 2ˣ. The solution is the x value of S. Go to M4.10.8. Try another.

    2. Not this one. You solved 2x = 20. In 2ˣ the x is a power: 2¹⁰ is 1 024, far above 20. Go to M4.10.8. Try another.

    3. Yes, that is right. Go straight down from S to the x axis. The reading is between 4 and 4.5, about 4.3. Check: 2⁴ = 16 and 2⁵ = 32, and 20 lies between them.

    See the answer

    4.3

    Go straight down from S to the x axis. The reading is between 4 and 4.5, about 4.3. Check: 2⁴ = 16 and 2⁵ = 32, and 20 lies between them.

    • If you chose A: The number 20 is the height of the flat line, the value of 2ˣ. The solution is the x value of S. Go to M4.10.8.
    • If you chose B: You solved 2x = 20. In 2ˣ the x is a power: 2¹⁰ is 1 024, far above 20. Go to M4.10.8.

  5. The graphs of y = f(x) and y = g(x) cross at a point. What does the x value of that point tell you?

    Tap your answer.

    1. Not this one. The x value is read along the bottom axis. The height of the point is the y value, a different number. Go to M4.10.8. Try another.

    2. Not this one. That would be where the graph of f meets the x axis. At a crossing of two graphs, f(x) and g(x) are equal, but not necessarily 0. Go to M4.10.8. Try another.

    3. Not this one. The gradient is how steep a graph is, and the two graphs usually have different gradients where they cross. Go to M4.10.8. Try another.

    4. Yes, that is right. At the crossing point both graphs have the same y value for that x. So f(x) = g(x) is true there, and x is a solution.

    See the answer

    It is a solution of the equation f(x) = g(x).

    At the crossing point both graphs have the same y value for that x. So f(x) = g(x) is true there, and x is a solution.

    • If you chose A: The x value is read along the bottom axis. The height of the point is the y value, a different number. Go to M4.10.8.
    • If you chose B: That would be where the graph of f meets the x axis. At a crossing of two graphs, f(x) and g(x) are equal, but not necessarily 0. Go to M4.10.8.
    • If you chose C: The gradient is how steep a graph is, and the two graphs usually have different gradients where they cross. Go to M4.10.8.

  6. f(x) = 2/x. We find f(−2) = −1 and f(4) = 0.5. Does f have a root between −2 and 4?

    Tap your answer.

    1. Not this one. A change of sign shows a root only if the graph has no break. The graph of 2/x has a break at zero. Go to M4.10.8. Try another.

    2. Yes, that is right. The two branches of 2/x never touch the x axis. The graph jumps from below the axis to above it across the break at zero. So the change of sign does not mean a root.

    3. Not this one. There is no value of 2/x at zero, because nothing can be divided by zero. The graph does not meet the y axis. Go to M4.10.3. Try another.

    See the answer

    No. There is a break in the graph at zero, and the graph never meets the x axis.

    The two branches of 2/x never touch the x axis. The graph jumps from below the axis to above it across the break at zero. So the change of sign does not mean a root.

    • If you chose A: A change of sign shows a root only if the graph has no break. The graph of 2/x has a break at zero. Go to M4.10.8.
    • If you chose C: There is no value of 2/x at zero, because nothing can be divided by zero. The graph does not meet the y axis. Go to M4.10.3.

  7. From an earlier chapter

    A tangent to a distance time curve passes through (1, 3) and (5, 19), with distance in m and time in s. What is the speed, in m/s?

    Tap your answer.

    1. Yes, that is right. The rise is 19 − 3 = 16 m and the run is 5 − 1 = 4 s. The gradient is 16 ÷ 4 = 4, so the speed is 4 m/s.

    2. Not this one. You used only the rise, 19 − 3. The gradient needs the rise divided by the run. Go to M4.10.7. Try another.

    3. Not this one. You divided the end distance 19 by the end time 5. The tangent does not start at the origin, so use the rise and the run between the two points. Go to M4.10.7. Try another.

    See the answer

    4 m/s

    The rise is 19 − 3 = 16 m and the run is 5 − 1 = 4 s. The gradient is 16 ÷ 4 = 4, so the speed is 4 m/s.

    • If you chose B: You used only the rise, 19 − 3. The gradient needs the rise divided by the run. Go to M4.10.7.
    • If you chose C: You divided the end distance 19 by the end time 5. The tangent does not start at the origin, so use the rise and the run between the two points. Go to M4.10.7.

  8. f(x) = x² − 7. Find f(2.8).

    See the answer

    0.84

    2.8² = 2.8 × 2.8 = 7.84. Then 7.84 − 7 = 0.84.

  9. x² = 30. The positive solution lies between two whole numbers. What is the smaller one?

    See the answer

    5

    5² = 25 is below 30 and 6² = 36 is above 30. So the solution is between 5 and 6, and the smaller number is 5.

Keep it

Copy this into your exercise book.

Solve f(x) = g(x): read the x values where the graphs cross. Roots: where f changes sign, if there is no break. Narrow the gap by trying values.

Draw: A line and a curve crossing at two points, with their x values marked on the x axis.

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.