Virginia SOL Math / Algebra II / Functions

A2.F.1

Analyze & Transform Function Families

What you'll learn

  • Identify parent functions and their families
  • Apply translations, reflections, and stretches
  • Write the equation of a transformed function

Words to know

Parent function —
the simplest form in a family, like y = x^2
Translation —
a slide left, right, up, or down
Reflection —
a flip over an axis
Stretch —
a vertical or horizontal scaling
Vertex form —
y = a(x - h)^2 + k for a parabola

Why it matters Recognizing how a graph shifts or stretches lets us predict its shape from an equation without plotting many points.

Worked examples

Model ItI do

Describe the graph of y = (x - 2)^2 + 1.

Start with the parent y = x^2, whose vertex is (0, 0). The '- 2' inside shifts it 2 units right, and the '+ 1' outside shifts it 1 unit up. The new vertex is (2, 1), and the parabola opens up.

Try It TogetherWe do

How does y = -x^2 differ from y = x^2?

The negative sign reflects the parabola over the x-axis, so it opens down instead of up. The vertex stays at (0, 0).

Now YouYou do

Describe the graph of y = (x + 3)^2 - 4.

Read the shifts from the vertex form and name the vertex. Check your answer in the back of this chapter.

Practice

Mastered 0 of 11

Warm-up

Ease in with the basics.

  1. 1.What is the vertex of y = (x - 6)^2 + 4?

  2. 2.Which way does y = x^2 - 5 shift from y = x^2?

  3. 3.Does y = -x^2 open up or down?

On Level

The core Quick Check from the book.

  1. 1.What is the vertex of y = (x - 5)^2 + 2?

  2. 2.Which way does y = x^2 + 3 shift from y = x^2?

  3. 3.Does y = -x^2 open up or down?

  4. 4.What is the vertex of y = (x + 1)^2?

  5. 5.Which way does y = (x - 4)^2 shift from y = x^2?

Challenge

Stretch your thinking.

  1. 1.Describe the transformations of y = -2(x + 3)^2 - 1 from y = x^2.

  2. 2.The graph of y = sqrt(x) is shifted 4 units left and 2 units up. Write the new equation.

  3. 3.What is the vertex of y = 3(x - 1)^2 + 7, and does the graph open up or down?