Question

Write a function of the form \(y=a b^{x-h}+k\) whose graph has a \(y\)-intercept of 5 and an asymptote of \(y=2\).

Short answer

The function is \(y = 3 * (1) ^ {x - 0} + 2\), which simplifies to \(y = 3 + 2 = 5\). To confirm, when x is 0, y is indeed 5, and as x approaches infinity, y approaches the asymptote of 2.

Step-by-step solution

Determine the constant 'k'

The constant 'k' is the value y approaches as x approaches infinity. Given that the asymptote of the function is \(y = 2\), the value of 'k' is 2.

Determine the constant 'a'

The term 'a' is the difference between the y-intercept and the asymptote. Given that the y-intercept is 5 and the asymptote (value of 'k') is 2, the value of 'a' is \(5 - 2 = 3\).

Determine the constant 'b'

Since 'h' is not given and there's no information to find it, we can assume 'h' to be 0. This makes the function become \(y = ab^x + k\). When x is 0, this equation becomes \(y = a + k\), which must be equal to the y-intercept. Since we know 'a' and 'k', we can set \(3 + 2 = 5\) and confirm 'b' does not affect the intercept. Therefore, 'b' can be any real number. We can assume 'b' to be 1 for simplicity.

Key concepts

Understanding the y-intercept in Exponential Functions

Exponential functions like the one given in the exercise have various parts, and one crucial part is the y-intercept.

The y-intercept is the value where the graph of the function crosses the y-axis. In other words, it represents the value of the function when the input, or \(x\), is 0. This is a handy value because it gives you a specific point to locate on the graph. If you're working with the graph described in the exercise, the y-intercept is 5, which means the point \((0, 5)\) is on the function's graph.

To find the y-intercept, you usually set \(x=0\) in the function equation. For function transformations such as \(y=a b^{x-h}+k\), setting \(x=0\) simplifies to \(y=a b^{-h}+k\), which becomes \(y=a+k\) because we assumed \(h=0\) for simplicity.

So, once you know the values of \(a\) and \(k\), those allow you to calculate the y-intercept without any hassle.

Exploring Asymptotes and Their Role

An asymptote is a line that the graph of a function approaches but never actually reaches. It's a guiding line that frames the function's behavior at the extremes.

In the exercise, the asymptote is a horizontal line at \(y=2\). This means that as \(x\) becomes very large, either positively or negatively, the values of the function \(y=a b^{x-h}+k\) approach 2.

To identify the asymptote in an exponential function like this one, look for the constant \(k\). The value of \(k\) essentially shifts the baseline of the function up or down. So, knowing the asymptote helps establish where the function flattens out on a graph. In our case, every exponential curve based on that function would get flatter and "trail" at the horizontal line \(y=2\). Asymptotes help you predict the end behavior of the graph without plotting every value.

Getting a Grip on Function Transformations

Function transformations involve altering the basic function to translate, stretch, shrink, or reflect it. These transformations allow you to modify the graph's appearance using the equation format \(y=a b^{x-h}+k\).

• **Vertical Shift**: Controlled by \(k\), it moves the graph up or down.
• **Horizontal Shift**: Managed by \(h\), it slides the graph left or right.
• **Vertical Stretch/Shrink**: Adjusted by \(a\), it changes how steep or flat the function appears.
• **Reflection**: If \(a\) is negative, the graph flips over the x-axis.

Understanding function transformations helps you visualize how changes in the mathematical formula affect the graph shape. For instance, in the provided solution, \(k=2\) determined the asymptote, \(a=3\) adjusted the graph’s steepness and is integral in ensuring the y-intercept is at 5.

These transformations make it possible to tailor functions to fit very precise real-world data, giving us immense flexibility in graph analysis. They're key for adapting the basic exponent function \(b^{x}\) into more complex models fitting specific conditions.