1.The graph of y = - 1/2 |x - 5| - 3 can be obtained from the graph of y = |x| by which transformations?

What is the horizontal shift?
By what factor is the graph stretched or shrunk vertically and how is it reflected?
What is the vertical shift?
2. Find the following for the function f(x) = (x + 6)^2 (x - 4)^2
(a) Find the x-and y-intercepts of the polynomial function f.
(b) Determine whether the graph of f crosses or touches the x-axis at each x-intercept
(c) Put all the information together to obtain the graph of f.
The x-intercept(s) is (are)?
The y-intercept of f is?
Does the graph cross or touch the x-axis at the smaller x-intercept?
Does the graph cross or touch the x-axis at the larger x-intercept?
What is the graph suppose to look like for this question?
Thanks for the help.

1. The graph of y = -1/2 |x - 5| - 3 can be obtained from the graph of y = |x| by the following transformations:

- Horizontal Shift: The graph is shifted 5 units to the right.
- Vertical Stretch/Shrink: The graph is vertically shrunk by a factor of 1/2.
- Reflection: The graph is reflected vertically because of the negative coefficient (-1/2).
- Vertical Shift: The graph is shifted downward 3 units from the original graph.

2. For the polynomial function f(x) = (x + 6)^2 (x - 4)^2:
(a) To find the x-intercepts, set f(x) = 0 and solve for x. The x-intercepts are the values where the graph crosses the x-axis.
(b) To determine if the graph crosses or touches the x-axis at each x-intercept, we need to check the multiplicity of each factor. If the multiplicity is even, the graph touches the x-axis, and if the multiplicity is odd, the graph crosses the x-axis.
(c) By analyzing the x-intercepts, y-intercept, and the information from (b), we can sketch the graph of f.

The x-intercepts are the values of x for which f(x) = 0.
The y-intercept of f is the value of f(0).
The graph either crosses or touches the x-axis at each x-intercept, depending on the multiplicity of the factor.
The graph should show a parabolic shape that touches or crosses the x-axis at the given x-intercepts.

Hope that helps! Feel free to ask if you have any more questions.

1. To obtain the graph of y = -1/2 |x - 5| - 3 from the graph of y = |x|, the following transformations are applied:

(a) Horizontal Shift: The graph of y = -1/2 |x - 5| - 3 is horizontally shifted 5 units to the right from the graph of y = |x|. This means the horizontal shift is +5.

(b) Vertical Stretch/Shrinking and Reflection: The graph y = |x| is reflected vertically because of the negative sign (-1/2) in front of the absolute value. Additionally, the absolute value function is vertically shrunk by a factor of 1/2 since the coefficient in front of the absolute value is less than 1.

(c) Vertical Shift: The graph y = -1/2 |x - 5| - 3 is vertically shifted downward by 3 units from the graph of y = -1/2 |x - 5|. This means the vertical shift is -3.

2. For the function f(x) = (x + 6)^2(x - 4)^2:

(a) X-Intercepts: To find the x-intercepts, set f(x) equal to zero and solve for x. So, (x + 6)^2(x - 4)^2 = 0. This equation has two solutions: x = -6 and x = 4. Therefore, the x-intercepts are -6 and 4.

Y-Intercept: To find the y-intercept, substitute x = 0 into the function f(x). So, f(0) = (0 + 6)^2(0 - 4)^2 = 36 * 16 = 576. Therefore, the y-intercept is 576.

(b) To determine if the graph of f crosses or touches the x-axis at each x-intercept, you need to consider the multiplicity of each factor. In the given function, both factors (x + 6) and (x - 4) are squared, which means their multiplicity is 2. This indicates that the graph of f touches but does not cross the x-axis at both x-intercepts (-6 and 4).

(c) Graph of f: The graph will touch the x-axis at both x-intercepts (-6 and 4) and will have a positive parabolic shape since the factors are squared. The graph will rise to positive infinity on the right side and negative infinity on the left side.

Thus, the graph should have two x-intercepts at -6 and 4, a y-intercept at 576, and touch but not cross the x-axis at both x-intercepts. It will also have a positive parabolic shape.