Questions: Graph the exponential function. f(x) = (5/2)^x Plot five points on the graph of the function, and also draw the asymptote. Then click on the graph-a-function button.

Graph the exponential function. f(x) = (5/2)^x Plot five points on the graph of the function, and also draw the asymptote. Then click on the graph-a-function button.
Transcript text: Graph the exponential function. \[ f(x)=\left(\frac{5}{2}\right)^{x} \] Plot five points on the graph of the function, and also draw the asymptote. Then click on the graph-a-function button.
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Solution

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Solution Steps

Step 1: Identify Parameters

The given function is $f(x) = 1(2.5)^x$ where:

  • $a = 1$, which scales the function vertically.
  • $b = 2.5$, the base of the exponential function, determining its growth or decay.
Step 2: Determine the Shape

The graph goes upwards because $a = 1$ is positive. The function grows exponentially as $x$ increases since $b = 2.5$ is greater than 1.

Step 3: Plot Key Points

The y-intercept occurs at $x = 0$, giving $f(0) = 1$. For $x = 1$, $f(1) = 2.5$. For $x = -1$, $f(-1) = 0.4$.

Step 4: Draw the Graph

Connect the plotted points smoothly, keeping in mind the exponential growth or decay. The graph should approach the x-axis as an asymptote but never touch it.

Step 6: Label Points

Label at least five points on the graph to provide a clear representation of the function's behavior. Key points include $f(0) = 1$, $f(1) = 2.5$, and $f(-1) = 0.4$.

Final Answer:

The exponential function $f(x) = 1(2.5)^x$ is plotted with key points and the general behavior based on the parameters $a = 1$ and $b = 2.5$. The graph direction is upwards and it grows exponentially as $x$ increases.

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