Quadratic & Exponential Functions

What you'll learn

Identify quadratic and exponential functions from equations and graphs
Describe key features such as vertex, intercepts, and growth
Compare quadratic and exponential growth

Words to know

Quadratic function —: a function of the form y = ax^2 + bx + c
Exponential function —: a function of the form y = a * b^x
Vertex —: the highest or lowest point of a parabola
Axis of symmetry —: the vertical line that splits a parabola in half
Growth factor —: the base b in an exponential function
Parabola —: the U-shaped graph of a quadratic function

Why it matters Quadratics model arcs and areas while exponentials model populations, interest, and anything that multiplies, so telling them apart shapes real predictions.

Worked examples

Model ItI do

Compare the graphs of y = x^2 and y = 2^x.

Both curve upward, but the exponential y = 2^x eventually rises far faster than the quadratic y = x^2, passing it beyond x = 4.

Try It TogetherWe do

Find the vertex of y = x^2 - 4.

There is no x-term, so the axis of symmetry is x = 0. At x = 0, y = -4, so the vertex is (0, -4).

Now YouYou do

A population doubles each year starting at 100. Write a function for the population after x years.

Use y = a * b^x with the starting value and growth factor. Check your answer in the back of this chapter.

Practice

Mastered 0 of 11

Warm-up

Ease in with the basics.

1.Is y = 4^x quadratic or exponential?
2.Find the vertex of y = x^2 + 3.
3.In y = 7 * 3^x, what is the growth factor?

On Level

The core Quick Check from the book.

1.Is y = 3^x quadratic or exponential?
2.Find the vertex of y = x^2.
3.In y = 5 * 2^x, what is the growth factor?
4.Does y = x^2 open up or down?
5.Evaluate y = 2^x when x = 3.

Challenge

Stretch your thinking.

1.Find the axis of symmetry of y = x^2 - 6x + 5.
2.A quantity starts at 200 and triples each year. Write a function for the amount after x years.
3.Evaluate y = 3 * 2^x when x = 4.