QuestionDraw the graph of $f(x)=-4^{x}$

Studdy Solution

STEP 1

1. The function $f(x) = -4^x$ is an exponential function with a negative coefficient.
2. The base of the exponential function is 4, which is greater than 1.
3. The graph will reflect the properties of exponential functions with a negative sign affecting the output.

STEP 2

1. Understand the basic properties of the function $f(x) = 4^x$ .
2. Apply the negative sign to understand the transformation to $f(x) = -4^x$ .
3. Plot key points to sketch the graph.
4. Draw the graph based on the points and transformations.

STEP 3

Understand the basic properties of $f(x) = 4^x$ : - The function $4^x$ is an exponential growth function. - As $x$ increases, $4^x$ increases rapidly. - The graph passes through the point $(0, 1)$ because $4^0 = 1$ . - As $x$ decreases, $4^x$ approaches zero but never reaches it.

STEP 4

Apply the negative sign to transform $f(x) = 4^x$ to $f(x) = -4^x$ : - The negative sign reflects the graph of $4^x$ across the x-axis. - The graph of $-4^x$ will pass through the point $(0, -1)$ because $-4^0 = -1$ . - As $x$ increases, $-4^x$ decreases rapidly, becoming more negative. - As $x$ decreases, $-4^x$ approaches zero from the negative side.

STEP 5

Plot key points to sketch the graph: - Plot the point $(0, -1)$ . - Choose a few other values for $x$ to calculate corresponding $y$ -values: - For $x = 1$ , $f(1) = -4^1 = -4$ . - For $x = -1$ , $f(-1) = -4^{-1} = -\frac{1}{4}$ . - Plot these points: $(1, -4)$ and $(-1, -\frac{1}{4})$ .

STEP 6

Draw the graph based on the points and transformations: - Connect the plotted points with a smooth curve. - Ensure the curve reflects the x-axis. - As $x \to \infty$ , the curve should approach negative infinity. - As $x \to -\infty$ , the curve should approach zero from below.
The graph of $f(x) = -4^x$ is a reflection of the exponential growth function $4^x$ across the x-axis, passing through $(0, -1)$ and approaching zero from below as $x$ decreases.

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