Math / Algebra

QuestionGraph the exponential function. $f(x)=-2\left(\frac{1}{2}\right)^{x}$
Plot five points on the graph of the function, and also draw the asymptote. Then click on the graph-a-function button.

Studdy Solution

STEP 1

1. The function to graph is $f(x) = -2\left(\frac{1}{2}\right)^{x}$ .
2. We need to plot five points on the graph.
3. We need to identify and draw the horizontal asymptote.

STEP 2

1. Identify the general behavior of the function.
2. Calculate and plot five points on the graph.
3. Determine and draw the horizontal asymptote.
4. Graph the function using the plotted points and asymptote.

STEP 3

The function $f(x) = -2\left(\frac{1}{2}\right)^{x}$ is an exponential function with a base of $\frac{1}{2}$ , which means it is a decreasing function. The negative sign indicates a reflection over the x-axis.

STEP 4

Calculate five points by substituting different values of $x$ into the function:
- For $x = -2$ : $ f(-2) = -2\left(\frac{1}{2}\right)^{-2} = -2 \times 4 = -8 \] Point: $ (-2, -8) $
- For $x = -1$ : $ f(-1) = -2\left(\frac{1}{2}\right)^{-1} = -2 \times 2 = -4 \] Point: $ (-1, -4) $
- For $x = 0$ : $ f(0) = -2\left(\frac{1}{2}\right)^{0} = -2 \times 1 = -2 \] Point: $ (0, -2) $
- For $x = 1$ : $ f(1) = -2\left(\frac{1}{2}\right)^{1} = -2 \times \frac{1}{2} = -1 \] Point: $ (1, -1) $
- For $x = 2$ : $ f(2) = -2\left(\frac{1}{2}\right)^{2} = -2 \times \frac{1}{4} = -\frac{1}{2} \] Point: $ (2, -\frac{1}{2}) $

STEP 5

The horizontal asymptote of the function is $y = 0$ because as $x$ approaches infinity, $\left(\frac{1}{2}\right)^{x}$ approaches zero, and thus $f(x)$ approaches zero.

STEP 6

Plot the points $(-2, -8)$ , $(-1, -4)$ , $(0, -2)$ , $(1, -1)$ , and $(2, -\frac{1}{2})$ on the graph. Draw the horizontal asymptote $y = 0$ . Connect the points smoothly to represent the exponential decay and reflection over the x-axis.

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QuestionGraph the exponential function. $f(x)=-2\left(\frac{1}{2}\right)^{x}$
Plot five points on the graph of the function, and also draw the asymptote. Then click on the graph-a-function button.

Studdy Solution

STEP 1

1. The function to graph is $f(x) = -2\left(\frac{1}{2}\right)^{x}$ .
2. We need to plot five points on the graph.
3. We need to identify and draw the horizontal asymptote.

STEP 2

1. Identify the general behavior of the function.
2. Calculate and plot five points on the graph.
3. Determine and draw the horizontal asymptote.
4. Graph the function using the plotted points and asymptote.

STEP 3

The function $f(x) = -2\left(\frac{1}{2}\right)^{x}$ is an exponential function with a base of $\frac{1}{2}$ , which means it is a decreasing function. The negative sign indicates a reflection over the x-axis.

STEP 4

STEP 5

The horizontal asymptote of the function is $y = 0$ because as $x$ approaches infinity, $\left(\frac{1}{2}\right)^{x}$ approaches zero, and thus $f(x)$ approaches zero.

STEP 6

Plot the points $(-2, -8)$ , $(-1, -4)$ , $(0, -2)$ , $(1, -1)$ , and $(2, -\frac{1}{2})$ on the graph. Draw the horizontal asymptote $y = 0$ . Connect the points smoothly to represent the exponential decay and reflection over the x-axis.

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QuestionGraph the exponential function. f(x)=−2(12)xf(x)=-2\left(\frac{1}{2}\right)^{x}f(x)=−2(21​)xPlot five points on the graph of the function, and also draw the asymptote. Then click on the graph-a-function button.

Studdy Solution

STEP 1

1. The function to graph is f(x)=−2(12)x f(x) = -2\left(\frac{1}{2}\right)^{x} f(x)=−2(21​)x.2. We need to plot five points on the graph.3. We need to identify and draw the horizontal asymptote.

STEP 2

1. Identify the general behavior of the function.2. Calculate and plot five points on the graph.3. Determine and draw the horizontal asymptote.4. Graph the function using the plotted points and asymptote.

STEP 3

The function f(x)=−2(12)x f(x) = -2\left(\frac{1}{2}\right)^{x} f(x)=−2(21​)x is an exponential function with a base of 12 \frac{1}{2} 21​, which means it is a decreasing function. The negative sign indicates a reflection over the x-axis.

STEP 4

STEP 5

The horizontal asymptote of the function is y=0 y = 0 y=0 because as x x x approaches infinity, (12)x \left(\frac{1}{2}\right)^{x} (21​)x approaches zero, and thus f(x) f(x) f(x) approaches zero.

STEP 6

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QuestionGraph the exponential function. $f(x)=-2\left(\frac{1}{2}\right)^{x}$
Plot five points on the graph of the function, and also draw the asymptote. Then click on the graph-a-function button.

1. The function to graph is $f(x) = -2\left(\frac{1}{2}\right)^{x}$ .
2. We need to plot five points on the graph.
3. We need to identify and draw the horizontal asymptote.

1. Identify the general behavior of the function.
2. Calculate and plot five points on the graph.
3. Determine and draw the horizontal asymptote.
4. Graph the function using the plotted points and asymptote.

The function $f(x) = -2\left(\frac{1}{2}\right)^{x}$ is an exponential function with a base of $\frac{1}{2}$ , which means it is a decreasing function. The negative sign indicates a reflection over the x-axis.

The horizontal asymptote of the function is $y = 0$ because as $x$ approaches infinity, $\left(\frac{1}{2}\right)^{x}$ approaches zero, and thus $f(x)$ approaches zero.