About This Quiz
Concept: Function Transformations
Function transformations involve shifting, stretching, or compressing a function along the x-axis or y-axis. The primary focus here is on horizontal translations, which shift the graph of a function left or right without changing its shape.
Consider a quadratic function \\([latex]f(x) = ax^2 + bx + c[/latex]\\). To transform this function horizontally, we can define a new function \\([latex]g(x) = f(x - h)[/latex]\\), where \\(h\\) is the horizontal shift. If \\(h > 0\\), the graph of \\(g(x)\\) is shifted to the right by \\(h\\) units; if \\(h < 0\\), it is shifted to the left by \\(|h|\\) units.
To find the vertex of the transformed function, first rewrite the original function in vertex form \\([latex]f(x) = a(x - h_0)^2 + k_0[/latex]\\), where \\((h_0, k_0)\\) is the vertex of \\(f(x)\\). Then, apply the horizontal shift to find the new vertex of the transformed function \\(g(x) = f(x - h)\\).
Success Tips:
- Identify the Transformation Type: Determine whether the transformation involves a horizontal or vertical shift, stretch, or compression. In this quiz, the focus is on horizontal shifts.
- Vertex Form: Rewrite the given function in vertex form \\([latex]a(x - h_0)^2 + k_0[/latex]\\) to easily identify the vertex \\((h_0, k_0)\\) of the original function.
- Apply the Shift: If the function is transformed to \\([latex]g(x) = f(x - h)\\), adjust the vertex accordingly. If \\(h > 0\\), shift the vertex right by \\(h\\) units; if \\(h < 0\\), shift left by \\(|h|\\) units.
- Calculate the New Vertex: Substitute the new x-coordinate of the vertex back into the transformed function to find the corresponding y-value, which gives you the new vertex coordinates.
- Practice with Examples: Work through several examples to get comfortable with the process of transforming functions and finding their vertices. This will help you recognize patterns and solve problems more efficiently during the actual SAT.