Vertical and Horizontal Translations of Quadratic Functions
Grade 9 students in California Reveal Math Algebra 1 learn how vertical and horizontal translations shift a parabola without changing its shape. A vertical shift g(x)=x²+k moves the vertex up by k units when k>0 or down when k<0, changing only the y-coordinate. A horizontal shift g(x)=(x-h)² moves the vertex right when h>0 or left when h<0 — and the sign is tricky: (x+3)² means h=-3, a leftward shift. Combined, g(x)=(x-h)²+k places the vertex at (h,k). For example, g(x)=(x+2)²-7 has vertex at (-2,-7), shifted left 2 and down 7.
Key Concepts
Starting from the parent function $f(x) = x^2$, two types of translations shift the parabola without changing its shape:.
Vertical shift: $$g(x) = x^2 + k$$ moves the graph up $k$ units when $k 0$ and down $|k|$ units when $k < 0$. Only the $y$ coordinate of the vertex changes; the vertex moves from $(0, 0)$ to $(0, k)$.
Common Questions
What does g(x)=x²+k do to the graph of f(x)=x²?
Adding k outside the squared term shifts the entire parabola vertically. If k>0, the vertex moves up k units; if k<0, it moves down |k| units. The x-coordinate of the vertex stays 0.
What does g(x)=(x-h)² do to the parabola?
Subtracting h inside the squared term shifts the parabola horizontally. If h>0, the graph shifts right h units to vertex (h,0). If h<0, it shifts left, moving the vertex to (h,0).
Why is the horizontal shift direction counterintuitive?
g(x)=(x+3)² looks like a rightward shift, but rewriting it as (x-(-3))² reveals h=-3, so the graph actually shifts left 3 units to vertex (-3,0). Always rewrite in the form (x-h)² to read the direction correctly.
What is the vertex of g(x)=(x-h)²+k?
The vertex is (h,k). The combined form places the vertex at h units horizontally and k units vertically from the origin.
What is the vertex of g(x)=(x+2)²-7?
Rewriting: h=-2 and k=-7, so the vertex is (-2,-7). The graph is shifted left 2 and down 7 from the parent f(x)=x².
Which unit covers quadratic translations in Algebra 1?
This skill is from Unit 10: Quadratic Functions in California Reveal Math Algebra 1, Grade 9.