Math Problem Statement

Graph the function g(x) = |x| + 4 as a translation of the parent function f(x) shown. How did the transformation affect the domain and range? Reflect across the x-axis: f(x) = -(x^2 - 4). Reflect across the y-axis: f(x) = x^2 - 4. Graph each function as a vertical stretch or compression of the parent function. g(x) = 4.5|x|. g(x) = 0.5x^2.

Solution

Let's go over the math problem step by step based on what I can see in the image.

Question 1: Transformations of g(x)=x+4g(x) = |x| + 4

You are asked to graph the function g(x)=x+4g(x) = |x| + 4 as a transformation of the parent function f(x)=xf(x) = |x|. This is a vertical shift, and as written on the sheet, the transformation moves the graph 4 units up.

  • Domain: The domain remains the same as the parent function f(x)=xf(x) = |x|, which is all real numbers, or (,)(-\infty, \infty).
  • Range: Since the entire graph is shifted up by 4 units, the range starts at 4 and goes to infinity, or [4,)[4, \infty).

Question 2: Reflection across the x-axis

The problem asks you to reflect the function f(x)=x24f(x) = x^2 - 4 across the x-axis. Reflecting across the x-axis involves multiplying the function by 1-1, so the new equation becomes: f(x)=(x24)=x2+4.f(x) = -(x^2 - 4) = -x^2 + 4.

Question 3: Reflection across the y-axis

When reflecting a function across the y-axis, you replace xx with x-x. So, reflecting f(x)=x24f(x) = x^2 - 4 across the y-axis gives: f(x)=(x)24=x24.f(x) = (-x)^2 - 4 = x^2 - 4. Since x2x^2 is symmetric about the y-axis, the function remains the same.

Question 4: Vertical Stretch

For g(x)=4.5xg(x) = 4.5|x|, this represents a vertical stretch of the parent function f(x)=xf(x) = |x| by a factor of 4.5. The V-shape of the absolute value graph becomes steeper due to the vertical stretch.

Question 5: Vertical Compression

For g(x)=0.5x2g(x) = 0.5x^2, this is a vertical compression of the parent function f(x)=x2f(x) = x^2 by a factor of 0.5, making the parabola wider than the original.

Question 6: Shift

The function g(x)=3(x+2)2g(x) = 3(x + 2)^2 represents a parabola shifted 2 units to the left and vertically stretched by a factor of 3.

Question 7: Shift

For g(x)=(x5)2+2g(x) = -(x - 5)^2 + 2, the function is shifted 5 units to the right, flipped across the x-axis (because of the negative sign), and shifted 2 units upward.

Would you like further explanation on any of these steps? Here are five related questions to deepen your understanding:

  1. How would the graph of g(x)=x3g(x) = |x| - 3 compare to the parent function f(x)=xf(x) = |x|?
  2. What would the range be for the function g(x)=x+2g(x) = -|x| + 2?
  3. How does multiplying the function f(x)=x2f(x) = x^2 by a number greater than 1 affect its graph?
  4. What happens if you reflect f(x)=xf(x) = |x| over both the x-axis and the y-axis?
  5. How would you graph the function g(x)=x+3g(x) = -|x + 3|?

Tip: When transforming functions, it's helpful to sketch the parent function first and apply each transformation step by step to ensure accuracy.

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Math Problem Analysis

Mathematical Concepts

Algebra
Transformations of Functions
Reflections
Vertical Stretch and Compression

Formulas

g(x) = |x| + c (Vertical shift)
f(x) = -(x^2 - 4) (Reflection across the x-axis)
f(x) = (-x)^2 - 4 (Reflection across the y-axis)
g(x) = 4.5|x| (Vertical stretch)
g(x) = 0.5x^2 (Vertical compression)

Theorems

Properties of absolute value functions
Reflection properties
Transformation of quadratic functions

Suitable Grade Level

Grades 9-11