Reflections of Quadratic Functions Across Axes
Grade 9 students in California Reveal Math Algebra 1 learn how reflections transform quadratic functions across the x-axis and y-axis. Reflecting across the x-axis multiplies the entire function by -1: g(x)=-f(x)=-x², flipping the parabola downward so a negative leading coefficient a indicates this transformation. Reflecting across the y-axis replaces x with -x: g(x)=f(-x)=(-x)²=x², which looks identical for f(x)=x² but shifts the vertex when the function is not symmetric about the y-axis. In vertex form f(x)=a(x-h)²+k, reflecting across the y-axis changes the vertex from (h,k) to (-h,k).
Key Concepts
Given a quadratic function $f(x) = x^2$, reflections are defined as follows:.
Reflection across the x axis: Multiply the entire function by $ 1$:.
Common Questions
How do you reflect a quadratic function across the x-axis?
Multiply the entire function by -1. g(x)=-f(x) flips the parabola upside down. In vertex form, a negative leading coefficient a indicates an x-axis reflection.
How do you reflect a quadratic function across the y-axis?
Replace x with -x: g(x)=f(-x). For f(x)=(x-4)²+1, the reflection is g(x)=(-x-4)²+1=(x+4)²+1, shifting the vertex from (4,1) to (-4,1).
Why does reflecting f(x)=x² across the y-axis look the same?
Because x²=(-x)², the parabola is symmetric about the y-axis, so a y-axis reflection produces the identical graph. The effect becomes visible only when the vertex is off the y-axis.
What happens to the vertex when reflecting across the x-axis?
For f(x)=(x-3)²+2, reflecting across the x-axis gives g(x)=-(x-3)²-2. The vertex moves from (3,2) to (3,-2) and the parabola opens downward.
How does a negative a in vertex form relate to reflection?
In g(x)=-2(x+1)²+5, the negative coefficient a=-2 indicates a reflection across the x-axis combined with a vertical stretch by factor 2.
Which unit and grade level covers this skill?
This skill is from Unit 10: Quadratic Functions in California Reveal Math Algebra 1, Grade 9.