Vertical Stretches and Compressions of Quadratic Functions
Grade 9 students in California Reveal Math Algebra 1 learn how the coefficient a in g(x)=ax² controls whether a parabola is narrower or wider than the parent f(x)=x². When |a|>1 the parabola is vertically stretched — narrower than the parent — because output values grow faster. When 0<|a|<1 the parabola is vertically compressed — wider than the parent — because output values grow more slowly. For example, g(x)=3x² is narrower (stretch by 3), g(x)=(1/4)x² is wider (compress by 1/4), and g(x)=-2x² is narrower with a reflection (stretch by 2, reflected downward).
Key Concepts
Given the parent quadratic function $f(x) = x^2$, the transformed function is:.
$$g(x) = ax^2$$.
Common Questions
What does the coefficient a do in g(x)=ax²?
The coefficient a controls the width of the parabola. If |a|>1, the parabola is vertically stretched and appears narrower than the parent. If 0<|a|<1, it is compressed and appears wider.
Is g(x)=3x² narrower or wider than f(x)=x²?
g(x)=3x² is narrower. Since a=3>1, the graph is vertically stretched by a factor of 3, making it steeper than the parent parabola.
Is g(x)=(1/4)x² narrower or wider than the parent?
g(x)=(1/4)x² is wider. Since 0<a=1/4<1, the graph is vertically compressed, making it flatter than f(x)=x².
What happens when a is negative in g(x)=ax²?
A negative a both stretches or compresses the parabola and reflects it across the x-axis, so the parabola opens downward. For g(x)=-2x², the parabola is vertically stretched by 2 and points downward.
Does vertical stretch or compression change the vertex?
No. The vertex stays at the origin (0,0) for pure vertical stretches and compressions of the parent f(x)=x². Only the width and direction change.
Which unit covers vertical stretches and compressions of parabolas?
This skill is from Unit 10: Quadratic Functions in California Reveal Math Algebra 1, Grade 9.