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Math  /  Algebra

QuestionAll Bookr
Graph the function and its reflection about the yy-axis on the same axes, and give the yy-intercept. f(x)=5(14)xf(x)=5\left(\frac{1}{4}\right)^{x}
Clear All Draw: □\square
The yy-intercept is □\square . (Enter an ordered pair.)

Studdy Solution

STEP 1

1. We are given the function f(x)=5(14)x f(x) = 5\left(\frac{1}{4}\right)^{x} .
2. We need to graph this function and its reflection about the y y -axis.
3. We need to find the y y -intercept of the function.

STEP 2

1. Graph the function f(x)=5(14)x f(x) = 5\left(\frac{1}{4}\right)^{x} .
2. Reflect the function about the y y -axis.
3. Find the y y -intercept of the function.

STEP 3

To graph the function f(x)=5(14)x f(x) = 5\left(\frac{1}{4}\right)^{x} , plot several points by choosing values for x x and calculating f(x) f(x) .
For example: - When x=0 x = 0 , f(0)=5(14)0=5 f(0) = 5\left(\frac{1}{4}\right)^{0} = 5 . - When x=1 x = 1 , f(1)=5(14)1=54 f(1) = 5\left(\frac{1}{4}\right)^{1} = \frac{5}{4} . - When x=−1 x = -1 , f(−1)=5(14)−1=20 f(-1) = 5\left(\frac{1}{4}\right)^{-1} = 20 .
Plot these points and draw a smooth curve through them.

STEP 4

To reflect the function about the y y -axis, replace x x with −x-x in the original function.
The reflected function is: f(−x)=5(14)−x=5(4)x f(-x) = 5\left(\frac{1}{4}\right)^{-x} = 5\left(4\right)^{x}
Plot this reflected function by choosing values for x x and calculating f(−x) f(-x) .
For example: - When x=0 x = 0 , f(−0)=5(4)0=5 f(-0) = 5\left(4\right)^{0} = 5 . - When x=1 x = 1 , f(−1)=5(4)1=20 f(-1) = 5\left(4\right)^{1} = 20 . - When x=−1 x = -1 , f(1)=5(4)−1=54 f(1) = 5\left(4\right)^{-1} = \frac{5}{4} .
Plot these points and draw a smooth curve through them.

STEP 5

The y y -intercept of a function is the point where the graph intersects the y y -axis. This occurs when x=0 x = 0 .
For the original function f(x)=5(14)x f(x) = 5\left(\frac{1}{4}\right)^{x} , when x=0 x = 0 , f(0)=5 f(0) = 5 .
Therefore, the y y -intercept is the ordered pair (0,5) (0, 5) .
The y y -intercept is:
(0,5) \boxed{(0, 5)}

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