Combining Multiple Transformations in Vertex Form
Grade 9 students in California Reveal Math Algebra 1 learn to read all transformations of a parabola simultaneously from vertex form f(x)=a(x-h)²+k. The parameter a controls vertical stretch (|a|>1) or compression (0<|a|<1) and reflection across the x-axis if negative; h controls horizontal shift (right if h>0, left if h<0); and k controls vertical shift (up if k>0, down if k<0). The vertex is at (h,k). For example, f(x)=2(x-3)²+1 has vertex (3,1) — stretched by 2, shifted right 3, up 1; and f(x)=-1/2(x+4)²-2 has vertex (-4,-2) — compressed, reflected, shifted left 4, down 2.
Key Concepts
When a quadratic function is written in vertex form , all transformations are combined into a single equation:.
$$f(x) = a(x h)^2 + k$$.
Common Questions
What does each parameter in vertex form f(x)=a(x-h)²+k represent?
a controls vertical stretch/compression and reflection. h controls horizontal shift (right when positive, left when negative). k controls vertical shift (up when positive, down when negative). The vertex is at (h,k).
How do you find the vertex from vertex form?
The vertex is directly at (h,k). For f(x)=3(x-1)²-5, h=1 and k=-5, so the vertex is (1,-5).
What transformations does f(x)=2(x-3)²+1 describe?
Vertical stretch by 2, horizontal shift right 3 units, vertical shift up 1 unit. Vertex is at (3,1).
What transformations does f(x)=-1/2(x+4)²-2 describe?
Vertical compression by 1/2, reflection across the x-axis, horizontal shift left 4 units (since h=-4), vertical shift down 2 units. Vertex is at (-4,-2).
In what order are transformations applied in vertex form?
The stretch/compression and reflection from a are applied before the shifts from h and k. The shape of the parabola is set first, then the vertex position is moved.
Which unit covers combined transformations in vertex form?
This skill is from Unit 10: Quadratic Functions in California Reveal Math Algebra 1, Grade 9.