ScientistGrade 9⏱ 20 min

Functions and graphs

Feed numbers into a function machine and watch the table and graph grow: linear, quadratic and exponential.

Standards: HSF-IF.A.1HSF-IF.A.2HSF-IF.B.4

1Wonder

A taxi charges a fixed fee plus a price per kilometer. A ball thrown in the air rises and falls. Money in a savings account grows faster and faster. How can one idea describe all three?

2Explore

Function machine

This activity needs a bigger screen

Turn your phone sideways, or open this lesson on a tablet or computer. The explanation and quiz below work here.

Kind of function

Input x 2 Slope m 1 Intercept b 1

Try this

Move the input x. Watch the output, the table and the point on the graph move together.
In Linear, make the slope 0. What does the graph look like? What does the table show?
In Quadratic, change h and k. Where does the vertex go? When does the graph have no zeros?
In Exponential, try a growth factor below 1. What happens?

3Explain

Step 1 of 5

A function is a machine

A function takes an input, follows a rule, and gives exactly one output. With the rule f(x) = 2x + 1, an input of 3 always gives 7.

Three families

Linear $f(x) = mx + b$ : a straight line, the same change every step.
Quadratic $f(x) = a(x - h)^2 + k$ : a parabola with vertex $(h, k)$ .
Exponential $f(x) = a \cdot b^x$ : multiplies by the same factor $b$ every step.

4Practice

Question 1 of 5

If f(x) = 3x − 2, what is f(4)?

5Recap

A function gives exactly one output for each input.
Tables, graphs and rules are three ways to show the same function.
Key features: y-intercept, zeros, vertex (max/min), where it increases or decreases.
Vertical line test: if a vertical line can hit the graph twice, it is not a function.
Linear functions add the same amount each step; exponential functions multiply by the same factor.