Intercepts and Function Intervals

In this lesson:

• Find and interpret y-intercepts and x-intercepts in context
• Describe intervals where a function is increasing or decreasing
• Identify intervals where a function is positive or negative

What You Will Learn Today

1. Find and interpret the y-intercept as an initial value
2. Find x-intercepts (zeros) and explain their contextual meaning
3. Determine intervals where a function increases or decreases
4. Determine intervals where a function is positive or negative
5. Distinguish increasing/decreasing from positive/negative

What Do the Axis Crossings Mean?

• Where does the graph cross the y-axis? What does that mean?
• Where does it cross the x-axis? What does that mean?

Finding the crossing is algebra. Explaining it is the real goal.

The y-Intercept Gives the Initial Value

The y-intercept is where the graph crosses the y-axis — when .

• Point:
• Interpretation: Selling 0 items means $200 in losses — fixed costs before any sales

Profit Graph with Intercepts Labeled

• y-intercept at : loss of $200 before any sales
• The line rises — profit improves with each item sold
• The x-intercept tells us where profit finally reaches zero

The x-Intercept Is the Zero Crossing

The x-intercept (or zero) is where .

• Point:
• Interpretation: At 40 items, profit = $0 — the break-even point
• Below 40: losing money; above 40: making money

Soccer Ball Has Two x-Intercepts

A soccer ball is kicked upward:

where = seconds, = height in meters.

Check: Interpret the Two Zeros

→ zeros at and

• : ball is kicked (ground level)
• : ball lands (ground level)

Flight time = 4 seconds — span between the two zeros

Transition: From WHERE to HOW

You know WHERE the function crosses the axes. Now: HOW does it behave in between?

• Going up or down? → Increasing or decreasing
• Above or below the x-axis? → Positive or negative

Two separate questions. Two separate answers.

Defining Increasing and Decreasing Intervals

• Increasing: output rises as input increases — the graph goes up
• Decreasing: output falls as input increases — the graph goes down

Focus: direction of change, not whether the output is large or small.

"Going up" = increasing. "Going down" = decreasing.

Temperature Graph: Increasing and Decreasing Intervals

• Increasing: approximately 6am to 3pm — temperature is rising
• Decreasing: approximately 3pm to 6am (next day) — temperature is falling

Defining Positive and Negative Intervals

• Positive: — graph is above the x-axis
• Negative: — graph is below the x-axis

Focus: sign of the output, not whether it's growing or shrinking.

Positive = above zero. Negative = below zero.

The Key Distinction: All Four Combinations

Temperature at 80°F and dropping: positive (above 0) AND decreasing (falling).

Profit Graph: All Four Features

• Increasing: — growing; Decreasing: — shrinking
• Negative: — losing money; Positive: — making money

Writing Intervals Using Input Notation

Use the input variable (not output values):

• Increasing on → profit grows months 0 to 7
• Negative on → profit below zero before month 3
• Positive on → profit above zero after month 3

Check: What Does "Decreasing" Mean Here?

Temperature is 25°C and falling by 3°C per hour:

• Positive? YES — 25°C is above 0
• Increasing? NO — it is falling
• Decreasing? YES — output is going down

Result: positive AND decreasing — sign and direction are independent.

Independent Practice: Three Interpretation Problems

1. — tank volume (gallons). Find and interpret both intercepts.

2. A function is positive on , negative on . What is ?

3. A graph rises from to , then falls. Give the increasing interval.

Answers: Check Your Practice Work

1. — tank starts at 800 gallons (y-int). — empty after 32 hours (x-int).

2. is an x-intercept — the function equals zero there.

3. Increasing on

Key Takeaways: Intercepts and Intervals

• y-intercept: — initial value in context
• x-intercepts: — zero output; always interpret contextually
• Increasing/decreasing: direction of change
• Positive/negative: sign of output

Watch out: Increasing positive — analyze separately.

Coming Up: Extremes and End Behavior

In Lesson 2:

• Relative maximums/minimums — peaks and valleys; turning points
• End behavior — what happens as ?
• Symmetry — mirror patterns in the graph
• Sketching — draw a graph from a verbal description