Question

What transformations of f(x) = x² are applied to the function g?

A. g(x) = 2(x + 1)²
B. g(x) = (x - 3)² + 5
C. g(x) = -x² - 6
D. g(x) = 4(x - 7)² - 9

Answer 1

The transformations of f(x) = x² in the function g(x) include horizontal shifts, vertical shifts, reflections, and stretching or compressing. Examples are g(x) = 2(x + 1)² showing a horizontal shift left and stretch, g(x) = (x - 3)² + 5 showing a right shift and upward shift, g(x) = -x² - 6 reflecting over the x-axis and shifting down, and g(x) = 4(x - 7)² - 9 showing a right shift, stretch, and downward shift.

Step-by-step explanation:

Let's analyze the transformations applied to f(x) = x² that result in the various forms of function g(x). When comparing the base function to the transformed functions, we will look for horizontal shifts, vertical shifts, reflections, and stretching or compressing.

• For g(x) = 2(x + 1)², there is a horizontal shift one unit to the left (because of (x + 1)), and a vertical stretch by a factor of 2.
• g(x) = (x - 3)² + 5 has a horizontal shift three units to the right (due to (x - 3)) and a vertical shift up five units.
• In g(x) = -x² - 6, the function is reflected over the x-axis (due to the negative sign in front of x²) and shifted down by 6 units.
• Finally, g(x) = 4(x - 7)² - 9 has a horizontal shift seven units to the right, a vertical stretch by a factor of 4, and a vertical shift down by 9 units.