Topic 1: Translating the graph of a parabola: Two steps
Problem 1: The graph of y = x^2 is translated 2 units to the right and 3 units down. Write the equation of the new parabola and sketch its graph.
Problem 2: Sketch the graph of y = x^2 shifted 1 unit left and 4 units up. Provide the equation of the transformed parabola.
Topic 2: Translating the graph of an absolute value function: One step
Problem 1: The graph of y = |x| is translated 5 units up. Write the equation of the new function and sketch its graph.
Problem 2: Sketch the graph of y = |x| shifted 3 units to the left. Provide the equation of the transformed absolute value function.
Topic 3: Translating the graph of an absolute value function: Two steps
Problem 1: The graph of y = |x| is translated 2 units right and 1 unit down. Write the equation of the new function and sketch its graph.
Problem 2: Sketch the graph of y = |x| shifted 4 units left and 2 units up. Provide the equation of the transformed function.
Topic 4: How the leading coefficient affects the graph of an absolute value function
Problem 1: Compare the graphs of y = 2|x| and y = (1/2)|x| to y = |x|. Describe how the leading coefficient affects the shape of the graph.
Problem 2: Explain the effect of the leading coefficient in y = -3|x| compared to y = |x|. Sketch both graphs to illustrate the differences.
Topic 5: Translating the graph of a function: Two steps
Problem 1: The graph of f(x) = x^3 is translated 3 units to the left and 2 units up. Write the equation of the transformed function and sketch its graph.
Problem 2: Sketch the graph of f(x) = √x shifted 1 unit right and 3 units down. Provide the equation of the transformed function.
Topic 6: Transforming the graph of a function by reflecting over an axis
Problem 1: Reflect the graph of f(x) = x^2 over the x-axis. Write the equation of the new function and sketch its graph.
Problem 2: Reflect the graph of f(x) = x^3 over the y-axis. Provide the equation of the transformed function and sketch its graph.
Topic 7: Transforming the graph of a quadratic, cubic, square root, or absolute value function
Problem 1: Transform the graph of f(x) = √x by reflecting over the x-axis and shifting 2 units up. Write the equation and sketch the transformed graph.
Problem 2: For f(x) = x^2, apply a vertical stretch by a factor of 3 and a shift 1 unit left. Provide the equation and sketch the resulting graph.
Topic 8: How the leading coefficient affects the shape of a parabola
Problem 1: Compare the graphs of y = 3x^2, y = (1/3)x^2, and y = x^2. Describe how the leading coefficient affects the width of the parabola.
Problem 2: Explain the effect of the leading coefficient in y = -2x^2 compared to y = x^2. Sketch both parabolas to show the differences in shape.
Topic 9: Determining whether two functions are inverses of each other
Problem 1: Determine if f(x) = 2x + 3 and g(x) = (x - 3)/2 are inverses of each other. Compute (f ∘ g)(x) and (g ∘ f)(x) to verify.
Problem 2: Check if f(x) = x^3 and g(x) = ∛x are inverses. Show the composition of both (f ∘ g)(x) and (g ∘ f)(x) and confirm the result.