Lesson 4.9

Evaluating Functions Graphically

No equation? No problem. Use the map. Find your starting location and look for the destination.

Introduction

Sometimes we don't have a formula like , but we have a picture. We can still find inputs and outputs by tracing lines on the grid.

Past Knowledge

Lesson 4.1 (Coordinate Plane). Knowing X is horizontal and Y is vertical is everything here.

Today's Goal

Find by looking at a graph, and find when .

Future Success

Evaluating limits graphically () is the first week of Calculus I.

Key Concepts

The Two Directions

Find f(a)

Input Given → Find Output

Go to x = a on the X-axis.
Look Up or Down to hit the graph.
Look Over to the Y-axis to read the value.

Find x when f(x) = b

Output Given → Find Input

Go to y = b on the Y-axis.
Look Left or Right to hit the graph.
Look Down to the X-axis to read the value.

Worked Examples

Example 1: Finding Value

Basic

Use the graph to find .

Go to x=2

Find 2 on the Horizontal Axis.

Trace to Curve

Move up until you hit the blue line.

Read Y-value

The height is 4.

f(2) = 4

Example 2: Solving for X

Intermediate

Find all values of where .

Go to Y=3

Height is 3. Scan Left and Right.

Find Itersections

We hit the graph at TWO places!

One is at . The other is at .

x = -1, x = 3

Example 3: Zeros (Roots)

Advanced

Find all values where .

Ground Level

means height is zero. This is the X-axis.

Identify Intercepts

The graph crosses at -1, 2, and 4.

x = -1, 2, 4

Common Pitfalls

Axis Confusion

If asked for , find 2 on the X-axis. If asked for , find 2 on the Y-axis. The position of the 2 matters.

Writing Coordinates

If , the answer is just 5. Don't write the coordinate (3,5) unless asked for a "Point".

Real-Life Applications

Temperature Charts: Looking at a weather forecast. "What will the temperature be at 2 PM?" (Finding f(2)). "When will it hit 80 degrees?" (Solving f(x)=80).