Math Problem Statement
Solution
Let's go over the math problem step by step based on what I can see in the image.
Question 1: Transformations of
You are asked to graph the function as a transformation of the parent function . 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 , which is all real numbers, or .
- Range: Since the entire graph is shifted up by 4 units, the range starts at 4 and goes to infinity, or .
Question 2: Reflection across the x-axis
The problem asks you to reflect the function across the x-axis. Reflecting across the x-axis involves multiplying the function by , so the new equation becomes:
Question 3: Reflection across the y-axis
When reflecting a function across the y-axis, you replace with . So, reflecting across the y-axis gives: Since is symmetric about the y-axis, the function remains the same.
Question 4: Vertical Stretch
For , this represents a vertical stretch of the parent function 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 , this is a vertical compression of the parent function by a factor of 0.5, making the parabola wider than the original.
Question 6: Shift
The function represents a parabola shifted 2 units to the left and vertically stretched by a factor of 3.
Question 7: Shift
For , 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:
- How would the graph of compare to the parent function ?
- What would the range be for the function ?
- How does multiplying the function by a number greater than 1 affect its graph?
- What happens if you reflect over both the x-axis and the y-axis?
- How would you graph the function ?
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
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