Question

Describe the transformations that must be applied to the graph of each exponential function \(f(x)\) to obtain the transformed function. Write each transformed function in the form \(y=a(c)^{b(x-h)}+k\). a) \(f(x)=\left(\frac{1}{2}\right)^{x}, y=f(x-2)+1\) b) \(f(x)=5^{x}, y=-0.5 f(x-3)\) c) \(f(x)=\left(\frac{1}{4}\right)^{x}, y=-f(3 x)+1\) d) \(f(x)=4^{x}, y=2 f\left(-\frac{1}{3}(x-1)\right)-5\)

Short answer

a) \(y= \left(\frac{1}{2}\right)^{x-2} + 1\), b) \(y= -0.5 \, 5^{x-3}\), c) \( y= - \left( \frac{1}{4} \right)^{3x} + 1 \), d) \(y= 2 \, 4^{-\frac{x-1}{3}} - 5\)

Step-by-step solution

Identify the original function

For each sub-problem, start by writing down the given original exponential function.

Rewrite in standard form for a)

Given: \(f(x)=\left(\frac{1}{2}\right)^{x}, y=f(x-2)+1\).The transformations are a horizontal shift to the right by 2 units and a vertical shift up by 1 unit.The transformed function is \(y= \left(\frac{1}{2}\right)^{x-2} + 1\).

Rewrite in standard form for b)

Given: \(f(x)=5^{x}, y=-0.5 f(x-3)\).The transformations are a horizontal shift to the right by 3 units, a vertical stretch by a factor of 0.5, and a reflection across the y-axis.The transformed function is \(y= -0.5 \, 5^{x-3}\).

Rewrite in standard form for c)

Given: \(f(x)=\left(\frac{1}{4}\right)^{x}, y=-f(3 x)+1\).The transformations are a horizontal compression by a factor of \(\frac{1}{3}\), a reflection across the y-axis, and a vertical shift up by 1 unit.The transformed function is \( y= - \left( \frac{1}{4} \right)^{3x} + 1 \).

Rewrite in standard form for d)

Given: \(f(x)=4^{x}, y=2 f\left(-\frac{1}{3}(x-1)\right)-5\).The transformations are a horizontal shift to the right by 1 unit, a horizontal stretch by a factor of 3 (inverting the sign for a horizontal reflection), a vertical stretch by a factor of 2, and a vertical shift down by 5 units. The transformed function is \(y= 2 \, 4^{-\frac{x-1}{3}} - 5\).

Key concepts

Horizontal Shift

A horizontal shift in exponential functions involves moving the entire graph of the function left or right. The general form for this transformation is given by replacing each occurrence of the variable inside the function with another expression that includes added or subtracted constants.

For example, if you have the function \(f(x) = 3^x\) and you shift it to the right by 2 units, the transformed function will be \(g(x) = 3^{x-2}\).

This moves every point on the graph of \(f(x)\) 2 units to the right. Conversely, if the function is shifted left by 2 units, the transformation will look like \(g(x) = 3^{x+2}\).

Horizontal shifts do not affect the shape of the graph; they only change its position along the x-axis.

Vertical Shift

A vertical shift involves moving the graph of the exponential function up or down. The standard form for this transformation is achieved by adding or subtracting constants outside the function.

For instance, if we start with \(f(x) = 2^x\) and apply a vertical shift 3 units upward, the new function becomes \(g(x) = 2^x + 3\).

This moves every point on the graph of \(f(x)\) 3 units up, without altering the shape of the graph. Similarly, if we shift the function down by 3 units, we get \(g(x) = 2^x - 3\).

Vertical shifts are straightforward and only modify the y-values of the function.

Reflection

Reflections can occur across both the x-axis and y-axis. Reflecting an exponential function across the x-axis flips the graph upside down.

This is done by multiplying the entire function by -1. For example, \(f(x) = 2^x\) reflected across the x-axis becomes \(g(x) = -2^x\).

Reflecting across the y-axis involves changing the sign of the variable inside the function. The function \(f(x) = 2^x\) reflected across the y-axis becomes \(g(x) = 2^{-x}\).

Reflections change the orientation but maintain the overall shape of the graph.

Vertical Stretch/Compression

A vertical stretch or compression changes the steepness of the exponential graph. This is accomplished by multiplying the whole function by a positive constant.

If the constant is greater than 1, the graph stretches vertically. For example, if \(f(x) = 3^x\) becomes \(g(x) = 2 \times 3^x\), the graph will stretch vertically by a factor of 2.

If the constant is between 0 and 1, the graph compresses. For example, \(g(x) = 0.5 \times 3^x\) will compress the graph vertically by a factor of 0.5.

Vertical stretch/compression affects the y-values while keeping the x-values unchanged.

Horizontal Stretch/Compression

Horizontal stretch or compression involves changing the width of the graph. This transformation is applied by multiplying the variable inside the function by a positive constant.

If the constant is greater than 1, the graph compresses horizontally. For instance, \(f(x) = e^x\) transformed by \(g(x) = e^{2x}\) compresses horizontally by a factor of 2.

If the constant is between 0 and 1, the graph stretches horizontally. For example, \(g(x) = e^{0.5x}\) will stretch the graph horizontally by a factor of 2.

Horizontal stretch/compression affects the x-values, thus changing the width of the graph.