Grade by Grade
B. Exponential Functions · Exponential Functions

Transforming Exponential Graphs I: Translations, Vertical Stretches and Reflections

B2.2 B2.3About 75 minutes
Learning goals
  • I can describe how a, d and c in y = af(x − d) + c change the graph of f(x) = bˣ.
  • I can use the mapping rule (x, y) → (x + d, ay + c) to sketch a transformed graph from key points.
  • I can find the asymptote, domain, range and y-intercept of a transformed exponential function.
  • I can write the equation of a transformed exponential function from a description.

Words to know

transformation
A change to a graph: a translation (shift), reflection (flip) or stretch/compression.
vertical translation
Shifting a graph up or down. In y = f(x) + c the graph moves up c units (down if c < 0).
horizontal translation
Shifting a graph left or right. In y = f(x − d) the graph moves right d units (left if d < 0).
vertical stretch / compression
In y = af(x), every y-coordinate is multiplied by a: a stretch if |a| > 1, a compression if 0 < |a| < 1.
reflection in the x-axis
In y = −f(x) every y-coordinate changes sign, so the graph flips over the x-axis.
key points
A few easy points on the parent graph, such as (−1, 1/2), (0, 1), (1, 2), (2, 4) on y = 2ˣ, that you transform to sketch the new graph.

Moving and reshaping exponential graphs

In real models the plain graph y = 2ˣ almost never appears on its own. A cooling cup of tea levels off at room temperature, not at 0°C. A bank account starts at $500, not $1. To model these, we transform the parent graph: move it, flip it and stretch it.

The general transformed function is y = af(k(x − d)) + c. This lesson covers a, d and c. The next lesson adds k.

Two letters named a

The curriculum writes the parent function as f(x) = aˣ and the transformed function as y = af(k(x − d)) + c. To keep the two a's apart, this unit writes the base as b: the parent function is f(x) = bˣ, and a always means the vertical stretch.

Remember from Grade 10

You transformed y = x² into y = a(x − h)² + k: k moved the parabola up or down, h moved it right or left, and a stretched or reflected it. Exponential functions follow the same rules. (In the curriculum's notation for this course, the horizontal shift is called d and the vertical shift c.)

Investigation with technology

In Desmos, type y = a·2^(x − d) + c and add sliders for a, d and c. Move one slider at a time and watch:

  • c: the whole graph and its asymptote move up (c > 0) or down (c < 0). The asymptote becomes y = c.
  • d: the graph moves right when d > 0 and left when d < 0. (y = 2^(x − 3) moves right 3, because x must be 3 bigger to get the same output.) The asymptote does not change.
  • a: each y-value is multiplied by a. If |a| > 1 the graph stretches away from the x-axis; if 0 < |a| < 1 it is compressed toward it. If a < 0 the graph is reflected in the x-axis and lies below its asymptote.

If you do not have technology, the same discoveries come from tables of values, as in the examples below.

xy−4−3−2−10123−10123456789asymptote y = 3y = 2ˣ + 3y = 2ˣ
y = 2ˣ + 3 is y = 2ˣ (dashed) moved up 3 units. Every point rises 3, and so does the asymptote: it is now y = 3.
xy−3−2−10123−8−6−4−202468(0, 3)(1, 2)(1, 6)(1, −2)y = 3(2ˣ)y = 2ˣy = −2ˣ
y = 3(2ˣ) (long dashes) stretches y = 2ˣ vertically by a factor of 3: (1, 2) → (1, 6). y = −2ˣ (dotted) reflects it in the x-axis: (1, 2) → (1, −2). Neither changes the asymptote y = 0.

The mapping rule

For y = af(x − d) + c, every point (x, y) on the graph of f(x) = bˣ moves to

(x, y) → (x + d, ay + c)

and the asymptote y = 0 moves to y = c.

Order matters for the y-coordinate: multiply by a first (stretch or reflect), then add c (translate). The domain is always {x ∈ ℝ}. The range is {y ∈ ℝ | y > c} if a > 0, and {y ∈ ℝ | y < c} if a < 0.

Key point on y = 2ˣ(−1, 1/2)(0, 1)(1, 2)(2, 4)asymptote y = 0
y = 2ˣ + 3(−1, 3.5)(0, 4)(1, 5)(2, 7)y = 3
y = 2^(x − 4)(3, 1/2)(4, 1)(5, 2)(6, 4)y = 0
y = 3(2ˣ)(−1, 3/2)(0, 3)(1, 6)(2, 12)y = 0
y = −2ˣ(−1, −1/2)(0, −1)(1, −2)(2, −4)y = 0
Applying single transformations to the key points of y = 2ˣ.
Worked example

Example 1: One transformation at a time

Describe the transformation and give the asymptote and range: (a) y = 2ˣ − 5 (b) y = 2^(x + 2) (c) y = 0.5(2ˣ)

  1. (a) c = −5: translate down 5. Asymptote y = −5. Range {y ∈ ℝ | y > −5}.
  2. (b) x + 2 = x − (−2), so d = −2: translate left 2. Asymptote y = 0. Range {y ∈ ℝ | y > 0}. (The y-intercept is now 2² = 4.)
  3. (c) a = 0.5: vertical compression by a factor of 1/2 (each y-value halves). Asymptote y = 0, range y > 0, y-intercept 0.5.

(a) down 5; (b) left 2; (c) vertical compression by 1/2.

Worked example

Example 2: Sketching y = −3(2^(x + 1)) + 5

Describe the transformations of f(x) = 2ˣ, sketch the graph using key points, and state the domain, range, asymptote and y-intercept.

  1. Read the parameters: a = −3, d = −1, c = 5.
  2. Transformations: reflect in the x-axis and stretch vertically by 3 (a = −3); translate left 1 (d = −1); translate up 5 (c = 5).
  3. Mapping rule: (x, y) → (x − 1, −3y + 5).
  4. Key points: (−1, 1/2) → (−2, −1.5 + 5) = (−2, 3.5); (0, 1) → (−1, 2); (1, 2) → (0, −1); (2, 4) → (1, −7).
  5. Asymptote: y = 0 → y = 5. Since a < 0, the graph lies below the asymptote and decreases.
  6. Domain {x ∈ ℝ}; range {y ∈ ℝ | y < 5}. y-intercept: x = 0 gives −3(2¹) + 5 = −1 (matches the key point (0, −1)).

Reflected, stretched by 3, left 1, up 5; asymptote y = 5; range y < 5; y-intercept −1.

xy−4−3−2−10123−8−7−6−5−4−3−2−101234567(−2, 3.5)(−1, 2)(0, −1)(1, −7)asymptote y = 5y = 2ˣy = −3(2ˣ⁺¹) + 5
y = −3(2ˣ⁺¹) + 5 (solid) with the parent y = 2ˣ (dashed). The transformed key points are (−2, 3.5), (−1, 2), (0, −1) and (1, −7). The graph approaches the asymptote y = 5 from below as x → −∞.
Worked example

Example 3: A different base

Sketch y = 2(3^(x − 2)) − 1. State the domain, range and asymptote.

  1. Parent f(x) = 3ˣ with key points (−1, 1/3), (0, 1), (1, 3), (2, 9).
  2. a = 2 (vertical stretch by 2), d = 2 (right 2), c = −1 (down 1). Mapping: (x, y) → (x + 2, 2y − 1).
  3. (−1, 1/3) → (1, 2/3 − 1) = (1, −1/3); (0, 1) → (2, 1); (1, 3) → (3, 5); (2, 9) → (4, 17).
  4. Asymptote y = −1; a > 0, so the graph is above it. Domain {x ∈ ℝ}, range {y ∈ ℝ | y > −1}.

An increasing curve through (1, −1/3), (2, 1), (3, 5), approaching y = −1 on the left.

Just for fun

Why did y = 2ˣ + 3 never worry about hitting rock bottom?

Because somebody raised its floor to y = 3, and an asymptote is a floor you can never actually reach.

Hands-on activity

Make an exponential template

You need: grid paper, a sheet of tracing paper or baking parchment, a pencil and a ruler. (Like the parabola template from Grade 10, but for y = 2ˣ.)

  1. On grid paper, draw axes from x = −4 to 4 and y = −8 to 8 (1 square = 1 unit). Plot (−3, 1/8), (−2, 1/4), (−1, 1/2), (0, 1), (1, 2), (2, 4), (3, 8) and draw a smooth curve for y = 2ˣ. Draw the asymptote y = 0 as a dashed line.
  2. Lay the tracing paper on top and trace the curve, the key points and the dashed asymptote. This is your template.
  3. Slide the template (keep it straight) to show y = 2ˣ − 4 and then y = 2^(x − 2). Mark the new key points on the grid paper.
  4. Flip the template over the x-axis (turn it face down, top to bottom) to show y = −2ˣ. Where is the asymptote? Then slide the flipped template up 3 to show y = −2ˣ + 3.
  5. Explain why sliding cannot show y = 3(2ˣ). What must you do instead?

What to look for: y = 2ˣ − 4 has key point (0, −3) and asymptote y = −4; y = 2^(x − 2) has (2, 1) and asymptote y = 0. The flipped template shows y = −2ˣ below the x-axis through (0, −1); moved up 3 it passes through (0, 2) with asymptote y = 3 and lies below it. A vertical stretch changes the shape (each y-value is tripled), so it cannot be done by sliding: you must plot new points such as (1, 6) and (2, 12).

Your turn

1

Describe the transformation of f(x) = 2ˣ: y = 2ˣ + 6.

2

Describe the transformation of f(x) = 3ˣ: y = 3^(x − 5).

3

Describe the transformations: y = −4(2ˣ).

4

Write the mapping rule for y = 2(5^(x + 3)) − 7.

5

Find the image of (1, 2) on y = 2ˣ under y = 3(2^(x − 1)) + 4.

6

State the asymptote, domain and range of y = −(1/2)(4ˣ) + 2.

7

Find the y-intercept of y = 5(2^(x + 1)) − 3.

More practice: warm-up

1

Which parameter moves the asymptote?

2

y = 2^(x + 4) moves the graph of y = 2ˣ which way?

3

What is the asymptote of y = 3ˣ − 2?

4

Which transformation does a = −1 cause?

5

Does a vertical stretch change the asymptote y = 0?

More practice: on your own

1

Sketch y = 2^(x − 3) + 1 by listing four transformed key points from y = 2ˣ and the asymptote.

2

For y = −2(3ˣ) + 4, find the image of (1, 3) and of (−1, 1/3), the asymptote and the range.

3

The graph of y = 2ˣ is reflected in the x-axis, stretched vertically by 5, then translated 2 left and 6 up. Write the equation.

4

Does y = 3(2ˣ) − 12 have an x-intercept? If so, find it.

5

Explain why y = −2ˣ − 1 has no x-intercept.

Challenge

1

Show that y = 2^(x + 3) and y = 8(2ˣ) are the same function. What does this say about horizontal translations of exponential graphs?

2

An exponential graph has asymptote y = −3, lies below it, and passes through (0, −5). Its equation has the form y = a(2ˣ) + c. Find a and c.

3

For y = a(2^(x − d)) + c with a > 0, explain why the range never depends on a or d.

Check your understanding

Quiz

1.

What is the asymptote of y = 2^(x − 4) + 6?

2.

y = 3^(x + 2) is the graph of y = 3ˣ translated:

3.

Which equation is y = 2ˣ reflected in the x-axis?

4.

What is the range of y = −5(4ˣ) + 1?

5.

Under y = 2(3ˣ) − 4, the point (1, 3) maps to:

6.

What is the y-intercept of y = 4(2^(x − 1)) + 3?

For parents and teachers

Big idea: Expectations B2.2 (investigate the roles of the parameters with technology) and B2.3 (sketch transformed graphs and state domain and range). This lesson covers a, d and c; the next adds k.

How to help: Ask your teen to explain each parameter in one sentence ("c moves everything, including the asymptote, up or down"). If you have a phone or computer, Desmos (free) with sliders makes the investigation quick and visual.

Common difficulties:

  • Moving the wrong way for d: 2^(x − 3) moves right 3.
  • Forgetting to move the asymptote along with c, or moving it with a or d (only c moves it).
  • Adding c before multiplying by a. Stretch or reflect first, then translate.
  • Range: when a < 0 the range is y < c, not y > c.