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Graphs Of Functions

Graphs Of Functions

Reciprocal, exponential and mixed graphs

Shapes Core + Extended 8 min read

On this page
  1. In this note
  2. Formula bank
  3. Reciprocal graphs: which quadrants?
  4. Exponential graphs
  5. Mixed functions
  6. Ways they test it
  7. Mistakes to avoid
  8. Summary
  9. Practice
  10. Can I do this?
  11. Quiz

In this note

Syllabus 2.10/2.11: reciprocal, exponential and mixed graphs
  • 2.11Know which quadrants the branches of \(y=\dfrac ax\) lie in, and sketch it.
  • 2.10Complete tables of values for \(y=\dfrac ax\), \(y=a^{x}\), \(y=k\times a^{x}\) and mixed functions like \(y=x+\dfrac4x\), and draw them.
  • 2.10Use the graph to solve equations such as \(2^{x}=5\) and to find a minimum.

Before you start: Graph shapes. Shifted curves and asymptotes in full: Asymptotes. Next: Drawing graphs accurately.

Formula bank

ReciprocalLearn it\(y=\dfrac ax\) (so \(xy=a\))asymptotes: the x- and y-axes. \(a>0\): quadrants 1 and 3; \(a<0\): quadrants 2 and 4.Use it when a number divided by x.
ExponentialLearn it\(y=a^{x}\) (\(a>0\))passes through (0, 1); always above the x-axis.Use it when x is a power.
Exponential with a multiplierLearn it\(y=k\times a^{x}\)passes through (0, k).
Domain and rangeLearn it\(y=\dfrac ax\): \(x\ne0\), \(y\ne0\); \(y=a^{x}\): \(y>0\)read from the asymptotes.Use it when 'state the range'.

Reciprocal graphs: which quadrants?

\(y=\dfrac ax\) can't use \(x=0\) (dividing by zero), and \(y\) is never 0, so the graph has two separate branches that get closer and closer to the axes (the asymptotes) without touching them. The sign of \(a\) decides where the branches go.

y = 6/x-6-4-2246-6-4-2246xy
a > 0: quadrants 1 and 3 (top-right, bottom-left).
y = −6/x-6-4-2246-6-4-2246xy
a < 0: quadrants 2 and 4 (top-left, bottom-right).

General methodHow to draw or sketch \(y=\dfrac ax\)

  1. Draw the asymptotes: the x-axis and the y-axis.
  2. Decide the quadrants from the sign of a: test x = 1 (y = a). Positive → top-right (and bottom-left); negative → bottom-right (and top-left).
  3. Make a table with x on BOTH sides of 0 (e.g. −4, −2, −1, 1, 2, 4).
  4. Plot and draw two smooth branches, each getting closer to both axes but never touching or crossing them.
  5. Never join the two branches across x = 0.

Moved curves like \(y=\dfrac a{x-b}+c\) use the same rule around the point where their asymptotes cross - see Asymptotes.

Example Which quadrants does \(y=-\dfrac6x\) lie in? Give three points on it.

Test x = 1 and x = −1, then fill a table.

  1. \(x=1\): \(y=-6\) (quadrant 4); \(x=-1\): \(y=6\) (quadrant 2)
  2. \(x=2\): \(-3\); \(x=3\): \(-2\)

Answer quadrants 2 and 4; e.g. (1, −6), (2, −3), (3, −2)

Try it yourself

1
Which quadrants hold \(y=\dfrac8x\)? Type the two numbers.
Show answer

Answer 1, 3

2
\(y=\dfrac{12}x\) at \(x=8\)?
Show answer

Answer 1.5

3
\(y=\dfrac8x\) at \(x=-2\)?
Show answer

Answer −4

Exponential graphs

y = 2^x-3-2-11234-1123456789xy
y = 2ˣ: each step right doubles y.
x−2−10123
y0.250.51248

General methodHow to draw an exponential graph and use it

  1. Table: work out each value carefully; a negative power is a fraction (\(2^{-1}=0.5\)).
  2. Plot and draw a smooth curve; it gets closer and closer to the x-axis on one side but never touches it.
  3. To solve \(a^{x}=k\): go across from \(y=k\) to the curve, then down to the x-axis.

Example Use the graph of \(y=2^{x}\) to solve \(2^{x}=5\).

Across from 5 to the curve, then down.

  1. 5 is between 4 (x = 2) and 8 (x = 3), nearer 4

Answer \(x\approx2.3\)

Try it yourself

1
\(y=2^{x}-1\) at \(x=-1\)?
Show answer

Answer −0.5

2
\(y=3\times0.5^{x}\) at \(x=-1\)?
Show answer

Answer 6

Mixed functions

\(y=x+\dfrac4x\) mixes two shapes: for small x the \(\dfrac4x\) part is huge, for large x the \(x\) part takes over. In between there is a minimum.

y = x + 4/x12345612345678910xy
y = x + 4/x for x > 0: minimum at (2, 4).
x0.511.52345
y8.554.1744.3355.8

Ways they test it

Type 1 Complete the table

"Complete the table for \(y=\dfrac{x^{2}}2-\dfrac3x\)." (3 marks)

Spot it a table with gaps.

  1. Substitute each x carefully (brackets for negatives).
  2. Round as asked (often 2 d.p.).

For example x = 0.5

  1. \(\tfrac{0.25}2-6=-5.875\)

Answer −5.88

Type 2 Draw the graph

"On the grid, draw the graph of y = f(x) for −3 ≤ x ≤ 3, x ≠ 0." (4 marks)

Spot it 'x ≠ 0' means two branches.

  1. Plot every point.
  2. Smooth curves; never cross x = 0.

Type 3 Which quadrants?

"Sketch \(y=-\dfrac4x\)."

Spot it a negative reciprocal.

  1. Test x = 1 and x = −1.
  2. Two branches in quadrants 2 and 4.

Type 4 Use the graph

"Use your graph to solve \(3^{x}=6\)" or "find the minimum value of y".

Spot it after the drawing.

  1. Across from the y-value, then down.
  2. For a minimum, read the lowest point.

For example y = 3ˣ

  1. 6 is between 3 (x = 1) and 9 (x = 2)

Answer x ≈ 1.6

Mistakes to avoid

  • ✗ Joining the points with a ruler✓ a smooth curveThese graphs are curves.
  • ✗ \(y=\dfrac{12}x\) drawn through x = 0✓ two separate branchesIt is undefined at 0.
  • ✗ \(2^{-1}=-2\)✓ 0.5A negative power is a fraction.
  • ✗ \(y=-\dfrac6x\) in quadrants 1 and 3✓ quadrants 2 and 4Test x = 1: y is negative.

Summary

  • \(y=\dfrac ax\): asymptotes on the axes; \(a>0\) quadrants 1 and 3, \(a<0\) quadrants 2 and 4.
  • \(y=a^{x}\): through (0, 1), above the x-axis, multiplies by a each step.
  • Mixed functions have a turning point between the two behaviours.
  • Smooth curves; read solutions across and down.

If you see …

  • \(\dfrac ax\) with a negative abranches top-left and bottom-right.
  • "solve \(a^{x}=k\) using your graph"across from y = k, then down.

Practice

Practice: easy, medium, hard

Easy

1
\(y=\dfrac{12}x\) at \(x=5\)?
Show answer

Answer 2.4

2
\(y=2^{x}\) at \(x=-2\)?
Show answer

Answer 0.25

3
Which quadrants hold \(y=-\dfrac4x\)? Type the two numbers.
Show answer

Answer 2, 4

Medium

1
\(y=x+\dfrac4x\) at \(x=1.5\), to 2 d.p.?
Guide me

What is it asking? a value

What do we know? substitute

Plan \(1.5+\tfrac4{1.5}\)

Show working
  1. \(1.5+2.666\ldots=4.1666\ldots\)

Answer 4.17

2
\(y=\dfrac{x^{2}}2-\dfrac3x\) at \(x=0.5\), to 2 d.p.?
Guide me

What is it asking? a value

What do we know? \(\tfrac{0.25}2\)

Plan \(-\tfrac3{0.5}=-6\)

Show working
  1. \(0.125-6=-5.875\)
  2. −5.88 (2 d.p.)

Answer −5.88

3
\(y=3\times0.5^{x}\) at \(x=3\)?
Guide me

What is it asking? a value

What do we know? \(0.5^{3}=0.125\)

Plan × 3

Show working
  1. \(3\times0.125\)

Answer 0.375

Hard

1
\(y=\dfrac6x+x\). Work out y at x = 2.5 (to 1 d.p.) and say which of x = 2, 2.5, 3 gives the lowest y.
Guide me

What is it asking? a value and an x

What do we know? \(\tfrac6{2.5}+2.5\)

Plan compare with x = 2 (5) and x = 3 (5)

Show working
  1. \(2.4+2.5=4.9\)
  2. x = 2: 5; x = 3: 5

Answer 4.9 at x = 2.5

PartAnswer / workingMarksGuidance
4.9B1
x = 2.5B1
Total: 2 marks
2
Use \(y=3^{x}\) to estimate the solution of \(3^{x}=6\), to 1 d.p.
Guide me

What is it asking? x

What do we know? 6 is between 3 and 9

Plan across from y = 6 and down

Show working
  1. between x = 1 and x = 2
  2. \(3^{1.6}\approx5.8\), \(3^{1.7}\approx6.5\)
  3. x ≈ 1.6

Answer 1.6

PartAnswer / workingMarksGuidance
1.5 to 1.7B1
Total: 1 mark
3
\(y=\dfrac8x\): state the value x can never take and the value y can never take.
Guide me

What is it asking? two values

What do we know? vertical asymptote

Plan horizontal asymptote

Show working
  1. x = 0 divides by zero
  2. 8 ÷ x is never 0

Answer x ≠ 0, y ≠ 0

PartAnswer / workingMarksGuidance
bothB1
Total: 1 mark

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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