Question

Write a rule for \(g\) that represents the indicated transformations of the graph of \(f\). \(f(x)=e^x\); horizontal shrink by a factor of \(\frac{1}{2}\), followed by a translation 5 units up

Short answer

The rule for \(g\) that represents the transformations of the graph of the function \(f(x)=e^x\) is \(g(x)=e^{2x}+5\).

Step-by-step solution

Apply the Horizontal Shrink

To apply a horizontal shrink to the function \(f(x)\) by a factor of \(\frac{1}{2}\), replace every instance of \(x\) in \(f(x)\) with \(2x\). This gives the new function \(g(x) = e^{2x}\).

Apply the Vertical Translation

To translate the function \(g(x)\) 5 units up, add 5 to \(g(x)\). This gives the final transformed function \(g(x) = e^{2x} + 5\).

Key concepts

Horizontal Shrink

In mathematics, a horizontal shrink of a function involves squeezing the graph towards the y-axis. Imagine taking the sides of the graph and pushing them inward. When you carry out a horizontal shrink, you alter the x-values in the function. Technically, you replace every x in the function with a new value \(kx\), where \(k > 1\). This effectively makes the graph skinnier, while keeping the same height.
For instance, a horizontal shrink by a factor of \(\frac{1}{2}\) means swapping every \(x\) with \(2x\). This doubles the value inside the function, resulting in a narrower version of the original function. It's like having less room to spread out the graph horizontally.
It's crucial to remember that although the name suggests the graph is being made smaller, the x-values are actually increased with a multiplying factor greater than one.

Vertical Translation

Vertical translations move the graph up or down in a straight motion along the y-axis. This does not change the shape of the graph, only its position. When we say a function is moved vertically, it's about adding or subtracting a constant to the whole function.
If you translate a function upwards by 5 units, you simply add 5 to the function. Mathematically, for a given function \(g(x)\), you would write \(g(x) + 5\). Thus, every point on the graph moves the same number of units upwards, maintaining its original form.
Think of it as lifting or lowering the graph without altering its overall look. The curve remains identical; it simply sits higher or lower on the graph than before.

Exponential Function

An exponential function is a mathematical expression in the form \(f(x) = a^x\), where \(a\) is a positive constant raised to the power of the variable \(x\). Exponential functions showcase the characteristic of rapid growth or decay. When \(a > 1\), as in our case with \(e^x\), the function illustrates rapid increase, since \(e\) is the base of natural logarithms and approximates to 2.718.
The graph of an exponential function increases rapidly, starting from nearly zero. The steepness and direction depend on the base value we use. This makes them crucial in representing real-world phenomena such as population growth or radioactive decay where change happens exponentially.

• For exponential growth, the graph ascends rapidly.
• If modeling decay, as described by a factor between 0 and 1, it descends swiftly.

Their unique properties make exponential functions incredibly powerful in various fields like biology and finance, where understanding growth patterns can be essential.