Question

Sketch the graph of the function. $$ h(x)=e^{x-2} $$

Short answer

The graph of \( h(x)=e^{x-2} \) is the graph of \( e^x \) shifted 2 units to the right.

Step-by-step solution

Identify the base function

The base function here is \( e^x \). This is a well-known function whose shape should be familiar. The graph of \( e^x \) starts at (0,1) and increases quickly to the right. To the left, it approaches y=0.

Identify the transformation

The function \( h(x)=e^{x-2} \) is simply the function \( e^x \) translated to the right by 2 units. This transformation does not affect the shape of the graph, just its position.

Sketch the graph

To sketch the graph of \( h(x) \), begin by drawing the graph of \( e^x \), and then shift it to the right by two units. The y-intercept of the base function \( e^x \) is (0,1), so, after the translation, the y-intercept of the function \( h(x) \) will be (2,1). As before, the graph increases quickly to the right and approaches y=0 to the left.

Key concepts

Transformations of Functions

Understanding how basic function graphs can be transformed is crucial for graphing a wide variety of functions. In the case of the function h(x) = e^{x-2}, we need to look at the transformation that occurs to the base function e^x.

Imagine sliding the graph of e^x along the x-axis; this is exactly what happens when we subtract 2 from x. This horizontal shift is one of the most common transformations, and it's characterized by a change in the function's formula where x is replaced by (x-c), with c being the number of units of the shift.

A positive value of c shifts the graph to the right, while a negative value shifts it to the left. It's important to remember that these shifts do not change the overall shape of the graph; they simply reposition it along the axis.

Exponential Growth

Exponential functions represent rapid growth or decay and are recognizable by their distinctive 'J-shaped' curve. The function e^x, one of the most important exponential functions, describes continuous growth at an ever-increasing rate.

Exponential growth is present when the rate of change of a quantity is proportional to the initial amount. In other words, the bigger the quantity gets, the faster it grows. When graphing e^x, this results in a curve that starts out slowly from the y-intercept at (0,1) and then shoots upwards dramatically for positive values of x.

The transformed function h(x) = e^{x-2} still has this rapid growth characteristic; the shift to the right does not affect the 'growth factor' but simply moves the graph over on the x-axis while maintaining its original shape.

Sketching Graphs

Sketching the graph of an exponential function requires an understanding of its basic shape and how transformations alter its position. Starting with the base function e^x, we know that the graph passes through (0,1), grows rapidly as x increases, and never touches the x-axis, approaching it asymptotically as x becomes more negative.

The graph of h(x) = e^{x-2} is a horizontal shift of this base function. By understanding the properties of e^x, we can predict the behavior of h(x). The point to start sketching is the y-intercept, which has moved from (0,1) to (2,1) due to the shift. After plotting this new intercept, drawing the curve to match the rapid increase of the base function will give a good approximation of the graph.

Remember, a sketch doesn't need to be perfect - the goal is to represent the general shape and relative positions of key points like intercepts and asymptotes.