Graph transformations and parameter experiments | Year 10

01 / LEARN

The idea, step by step

Track each parameter

For $y = a b^{x - h} + k$ , $b > 0$ , $b \neq 1$ and $a \neq 0$ , $h$ shifts the graph horizontally and $k$ shifts it vertically. The horizontal asymptote becomes $y = k$ .

The size of $a$ stretches outputs from the asymptote; a negative $a$ reflects them across it. The $y$ -intercept must be calculated by substituting $x = 0$ , not guessed from one coefficient.

Make a controlled comparison

To test a conjecture, vary one parameter while holding the others fixed. Compare the same inputs and record which feature changes.

For example, comparing $2^{x}$ with $2^{x} + 3$ shows every output increasing by $3$ . The base remains $2$ , so the multiplicative structure relative to the asymptote is unchanged.

A common trap

$x - 2$ shifts a graph left by $2$ .

Instead: To reproduce an old input $u$ , the new input must be $x = u + 2$ . The graph shifts right.

02 / WORKED EXAMPLES

See the thinking

Example 1

Shift right and down

For $y = 2^{x - 1} - 3$ , find the horizontal asymptote and $y$ -intercept.

The vertical shift is $- 3$ , so the asymptote is $y = - 3$ .
At $x = 0$ , $y = 2^{- 1} - 3 = - 5 / 2$ .

y = - 3

; intercept

(0, - 5 / 2)

Example 2

Reflect and stretch

Compare $y = - 2 (3^{x}) + 1$ with $y = 3^{x}$ .

The factor $- 2$ reflects the outputs and stretches their size by $2$ .
The $+ 1$ shifts the result up, making asymptote $y = 1$ .
The range is $y < 1$ and the intercept is $(0, - 1)$ .

Reflection, vertical stretch by 2, then shift up 1.

Check your understanding

State the horizontal asymptote of $y = 4 (2^{x}) + 5$ .

Show answer

$y = 5$

The $4$ changes scale, while the $+ 5$ sets the asymptote.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the $y$ -intercept of $y = 2^{x} + 4$ .
2 marks
State the shift from $y = 3^{x}$ to $y = 3^{x + 2}$ .
2 marks
State the range of $y = 2^{x} - 7$ .
2 marks
Find the intercept and asymptote of $y = - 3 (2^{x}) + 6$ .
3 marks
Compare $y = 2^{x}$ with $y = 2^{x - 1}$ at $x = 0, 1, 2$ . Give the outputs of the second graph.
2 marks
Explain why $y = 2^{x} + 3$ does not have consecutive output ratio 2, even though its exponential part does.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the $y$ -intercept of $y = 2^{x} + 4$ .
2 marks
$2^{0} + 4 = 5$ .
$(0, 5)$
State the shift from $y = 3^{x}$ to $y = 3^{x + 2}$ .
2 marks
The inside change $x + 2$ gives an earlier input.
Left 2 units
State the range of $y = 2^{x} - 7$ .
2 marks
The exponential part is positive, so the result exceeds $- 7$ .
$y > - 7$
Find the intercept and asymptote of $y = - 3 (2^{x}) + 6$ .
3 marks
At zero, $- 3 + 6 = 3$ ; the vertical shift sets the asymptote.
$(0, 3)$ and $y = 6$
Compare $y = 2^{x}$ with $y = 2^{x - 1}$ at $x = 0, 1, 2$ . Give the outputs of the second graph.
2 marks
Substitute the stated inputs; each is half the original output.
$1 / 2, 1, 2$
Explain why $y = 2^{x} + 3$ does not have consecutive output ratio 2, even though its exponential part does.
3 marks
At $x = 0, 1, 2$ , outputs are $4, 5, 7$ , whose ratios are not constant. Distances above the asymptote $y = 3$ are $1, 2, 4$ .
The constant shift changes output ratios.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Integer and fractional index laws; connecting fractional powers with roots; simplifying surds; operations with surds; rationalising denominators; exact and approximate values; exponential tables and graphs; growth/decay factors; comparing linear/quadratic/exponential patterns; exponential equations; logarithms as inverses; logarithm laws; interpreting logarithmic scales; graph transformations and parameter experiments; choosing and evaluating models.

Official curriculum source ↗

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