Question

Match each function with the corresponding transformation of \(y=3^{x}\). a) \(y=2(3)^{x}\) b) \(y=3^{x-2}\) c) \(y=3^{x}+4\) d) \(y=3^{\frac{x}{5}}\) \(\mathbf{A}\) translation up \(\mathbf{B}\) horizontal stretch \(\mathbf{C}\) vertical stretch \(\mathbf{D}\) translation right

Short answer

a) C b) D c) A d) B

Step-by-step solution

Understand the Parent Function

The parent function is given as \( y = 3^x \).

Identify Vertical Stretch

For the function \( y = a \times 3^x \, a eq 1 \, a eq 0 \), the value of 'a' stretches the graph vertically. Given \( y = 2 \times 3^x \, a = 2 \), it is a vertical stretch by a factor of 2.

Identify Horizontal Shift - Right

For the function \( y = 3^{x - h} \, h > 0 \), the graph shifts to the right by 'h' units. Given \( y = 3^{x-2} \), there is a horizontal shift of 2 units to the right.

Identify Vertical Shift - Up

For the function \( y = 3^x + k \, k > 0 \), the graph shifts up by 'k' units. Given \( y = 3^x + 4 \), there is a vertical shift of 4 units up.

Identify Horizontal Stretch

For the function \( y = 3^{x/c} \, c > 1 \), the graph stretches horizontally by a factor of 'c'. Given \( y = 3^{x/5} \, c = 5 \), it is a horizontal stretch by a factor of 5.

Match Each Function to Its Transformation

a) \( y = 2 \times 3^x \, C \, vertical stretch \) b) \( y = 3^{x-2} \, D \, translation right \) c) \( y = 3^x + 4 \, A \, translation up \) d) \( y = 3^{x/5} \, B \, horizontal stretch \)

Key concepts

Vertical Stretch

When we talk about a vertical stretch, we are referring to the transformation of a function in such a way that it stretches away from the x-axis. The general form looks like this: \(y = a \times f(x)\). Here's what happens:
\(a > 1 \): The graph stretches vertically by a factor of 'a'.
\(0 < a < 1 \): The graph shrinks vertically by a factor of 'a'.
For example, with \(y = 2 \times 3^x\), the factor '2' stretches the graph upwards, making it taller and steeper.
In essence, all 'y' values get multiplied by '2'. So, the points on the graph move further away from the x-axis.
Think of this as pulling the graph upwards when 'a' is greater than 1.

Horizontal Shift

A horizontal shift moves the graph of a function left or right along the x-axis. The general form here is \(y = f(x - h)\). Here's how it affects the graph:
\(h > 0\): The graph shifts 'h' units to the right.
\(h < 0\): The graph shifts 'h' units to the left.
For instance, \(y = 3^{x-2}\) means the graph moves 2 units to the right since 'h' is positive.
All points on the graph slide to the right by these 2 units.
It's like shifting the entire graph sideways without changing its shape or orientation.

Vertical Shift

When a vertical shift occurs, the graph of the function moves up or down along the y-axis, without altering its overall shape. The form used here is \(y = f(x) + k\). Here's what it means:
\(k > 0\): The graph shifts 'k' units up.
\(k < 0\): The graph shifts 'k' units down.
Take \(y = 3^x + 4\) as an example. The '+4' means the graph moves up by 4 units.
All points on the graph go up these 4 units.
So, every 'y' value increases by 4, effectively lifting the entire graph upwards.

Horizontal Stretch

A horizontal stretch changes the width of the graph of a function. It makes the function wider or narrower. The general expression here is \(y = f(\frac{x}{c})\). Here's what happens:
\(c > 1\): The graph stretches horizontally by a factor of 'c'.
\(0 < c < 1\): The graph compresses horizontally by a factor of 'c'.
For \(y = 3^{x/5}\), 'c' is 5, stretching the graph horizontally.
This action makes the graph wider, as if you're pulling it outward from the y-axis.
In other words, it spreads out the graph along the x-axis.