1.Find the following for the function
f(x)=(x+6)^2(x-2)^2
a. what are the x and y-intercepts?
B. Does the graph cross or touch the x-axis?
c. Does the graph cross or touch the x-axis at the larger x-intercept?
2. Which of the following best describes how the graph of f can be obtained from the graph of y=1/x^2 f(x)= -2/x^2
choose correct answer
a. vertically stretched by a factor of -2
b. reflection over the y-axis, vertically shrink bby a factor 2
c. horizontal shift of 2 units to right
d. reflection over the y-axis, vertically stretch by a factor of 2
I think the answer is d--am I correct?
1.
a. (0,144),(-6,0),(2,0)
B. touch
c. touch
2.
(A), if you can have negative stretch. I'd have said stretch by 2, flip over x-axis.
x intercept when y = 0
that would be when x = -6 (bounces off) and when x = 2 (bounces off)
y intercept when x = 0
so when y = 6^2 + 2^2 = 40
sorry, 36 * 4 = 144
1. To find the x-intercepts, we set the function equal to zero and solve for x. The function is f(x) = (x+6)^2(x-2)^2. Setting f(x) equal to zero:
0 = (x+6)^2(x-2)^2
To solve this equation, we can use the zero-product property, which states that if the product of two or more factors is zero, then at least one of the factors must be zero. This means that either (x+6)^2 = 0 or (x-2)^2 = 0.
Solving (x+6)^2 = 0:
(x+6)^2 = 0
x+6 = 0
x = -6
Solving (x-2)^2 = 0:
(x-2)^2 = 0
x-2 = 0
x = 2
Therefore, the x-intercepts are -6 and 2.
To find the y-intercept, we substitute x = 0 into the function:
f(0) = (0+6)^2(0-2)^2
f(0) = 6^2(-2)^2
f(0) = 36 * 4
f(0) = 144
Therefore, the y-intercept is (0, 144).
b. To determine if the graph crosses or touches the x-axis, we need to locate the x-values of the x-intercepts (-6 and 2). By analyzing the equation/factors, we can determine that the graph touches the x-axis at both x = -6 and x = 2, rather than crossing it.
c. Since the x-intercept at x = 2 is larger than the one at x = -6, the graph touches the x-axis at the larger x-intercept (x = 2).
2. To understand how the graph of f(x) = -2/x^2 is related to y = 1/x^2, we need to identify the transformations involved.
The original function y = 1/x^2 represents a hyperbola with its vertex at the origin (0, 0). By multiplying the original function by -2, we vertically reflect the graph over the x-axis. Then, by replacing x with -x in the function, we horizontally reflect the graph over the y-axis.
Therefore, the correct answer would be d. The graph of f(x) = -2/x^2 is obtained by reflecting the graph of y = 1/x^2 over the y-axis and vertically stretching it by a factor of 2.