Evaluating and Interpreting Functions

Lesson 1 of 2: HSF.IF.A.2

In this lesson:

• Evaluate functions at numeric and expression inputs
• Interpret function notation statements in context

Learning Objectives for This Lesson

By the end, you will be able to:

1. Evaluate functions at numeric inputs, including negatives and fractions
2. Evaluate functions at expression inputs such as and
3. Interpret in terms of a real-world context
4. Read function values from tables and graphs

Quick Review: You Already Know This

Let . Find .

Today we go further:

• What is ? What is ?
• What does tell us about a real situation?

Evaluating Functions: The Core Idea

Replace every with the input — wrapped in parentheses.

• Check the domain before substituting
• Parentheses protect signs throughout every operation

at :

Evaluating a Quadratic at Several Inputs

Quick Check: Evaluate at a Negative Input

Let .

Find .

Write out the substitution with parentheses, then simplify.

Negative Input Evaluated: Result Is Twelve

Notice: , not — the parentheses make this happen.

Expression Inputs: Replace with the Whole Expression

When the input is an expression, substitute the entire expression for every :

Quadratic Function with Expression Inputs

Let . Evaluate:

Common Error:

Let .

, so .

is a rule, not a multiplier.

Your Turn: Evaluate an Expression Input

Let .

Find .

Show the substitution step, then expand.

Answer:

Practice Problems: Evaluate These Functions

Let .

1. Find
2. Find
3. Find and simplify

Pause and work each one before the next slide.

Answers to Function Evaluation Practice

Interpreting Function Notation: A New Question

Restatement (not enough): " means f of 3 equals 85"

Interpretation (correct): "Three hours after the pool opens, there are 85 swimmers"

Use the contextual meaning of input and output — name the units and quantities.

Pool Context: = Swimmers After Hours

The number of swimmers at a city pool hours after opening is .

• : "When the pool opens, 12 swimmers are already there"
• : "Three hours after opening, there are 85 swimmers"
• : "Eight hours after opening, no swimmers remain"

Reverse Direction: Context to Notation

Given the pool context , write function notation for:

• "Five hours after opening, there are 60 swimmers" →
• "The pool has the same attendance at 2 hours and 6 hours" →

Quick Check: Write an Interpretation

The pool context: = number of swimmers, hours after opening.

Interpret:

Write a complete sentence that names the input and output in context.

Interpreting Production Cost Function Values

= dollars to produce units.

Guided Practice: Write Your Interpretation

= altitude of a plane (in feet) after minutes of flight.

Interpret:

Write a complete sentence using the context: mention altitude, feet, and minutes.

Practice: Translate Notation to Context

1. = town population after years. Interpret .
2. = temperature after hours past midnight. Write notation for "At 3 AM: 42°F."
3. = population. What does mean?

Write complete sentences for each before the next slide.

Answers to Interpretation Practice Problems

1. Ten years after the start, the town's population is 4,500 people.
3. The town's population at 5 years equals its population at 15 years — it returned to the same size after a decade.

Key Takeaways from Lesson One

• Wrap input in parentheses — protects signs throughout
• Expression inputs: replace every with the whole expression
• — is a rule, not a multiplier
• Interpret with context and units — not just the notation
• Check the domain; evaluate from any representation

Next: Lesson 2 — Solving and Compound Notation

Coming up in Lesson 2:

• Solve — find the input that produces a given output
• Read function values from graphs and tables (all representations)
• Distinguish from

Standard: HSF.IF.A.2 continues