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HSF · LE.A.2

Constructing Functions from Graphs and Descriptions

Deck 2 of 2: Graphs, Words, and Mixed Representations

In this deck:

  • Extract two points from a graph to build the function
  • Translate verbal descriptions into linear or exponential functions
  • Classify and construct from mixed representations
Construct Linear and Exponential Functions · Slide {page}
HSF · LE.A.2

What You Will Learn Today

  1. Read two integer-coordinate points from a graph
  2. Construct linear and exponential functions from graphs
  3. Translate verbal descriptions into function parameters
  4. Classify and construct from mixed representations
Construct Linear and Exponential Functions · Slide {page}
HSF · LE.A.2

Reading a Graph: Choose Clean Points

From a graph, choose two points where the curve passes through integer intersections.

  • Avoid midpoints between grid lines — reading errors compound downstream
  • Check: does the curve clearly pass through this intersection?
  • Pick points far apart to reduce slope computation error
Construct Linear and Exponential Functions · Slide {page}
HSF · LE.A.2

Linear Graph: Construct from Two Points

Coordinate graph with a rising line passing through labeled points (1,5) and (4,11), slope triangle showing rise=6 and run=3

  • Points: and
  • ; then

— verify: ✓ and ✓

Construct Linear and Exponential Functions · Slide {page}
HSF · LE.A.2

Exponential Graph: Construct from Two Points

Coordinate graph with an exponential curve passing through labeled points (0,3) and (2,12), showing the rapid upward curve

  • Points: and
  • ; then

— verify: ✓ and ✓

Construct Linear and Exponential Functions · Slide {page}
HSF · LE.A.2

Check: Which Points Did You Read?

For a graph passing through and :

  1. Compute the slope.
  2. Find using one point.
  3. Write the function.
  4. Verify both points.

Classify first: are these differences or ratios — linear or exponential?

Construct Linear and Exponential Functions · Slide {page}
HSF · LE.A.2

Answer: f(x) = 3x + 2

  • ; using :
  • — verify: ✓ and ✓

Constant difference of 3 → linear confirmed.

Construct Linear and Exponential Functions · Slide {page}
HSF · LE.A.2

Verbal Descriptions: Extract the Parameters

Verbal cue Parameter
"starts at" / "at time zero" initial value or
"increases by X per unit" slope
"grows by X% per unit"
"decreases by X% per unit"

Classify first, then extract the two parameters.

Construct Linear and Exponential Functions · Slide {page}
HSF · LE.A.2

Linear from Words: Pool Water Level

"A pool contains 200 gallons at . Water fills at 50 gallons per minute."

  • Initial value: →
  • Rate: 50 gallons/min →
  • Function:
Construct Linear and Exponential Functions · Slide {page}
HSF · LE.A.2

Exponential from Words: Car Depreciation

"A car costs 25,000 dollars new and depreciates 18% per year."

  • Initial value: →
  • Rate: −18%/yr → factor
  • Function:

⚠️ The factor is 0.82, not 0.18.

Construct Linear and Exponential Functions · Slide {page}
HSF · LE.A.2

Check: Write the Function from Words

"A salary starts at 45,000 dollars and increases by 2,500 dollars per year."

  1. Is this linear or exponential? How do you know?
  2. Identify and .
  3. Write .
  4. What is the salary after 5 years?

Work through all four steps before the next slide.

Construct Linear and Exponential Functions · Slide {page}
HSF · LE.A.2

Answer: S(t) = 2500t + 45000

  • Linear: constant dollar increase per year
  • ,
  • (57,500 dollars)
Construct Linear and Exponential Functions · Slide {page}
HSF · LE.A.2

Transition: From Single to Mixed Representations

You now have four construction pathways:

  • Two points (numbers)
  • Table of values
  • Graph
  • Verbal description

The underlying algebra is always the same. The only new step: classify first.

Construct Linear and Exponential Functions · Slide {page}
HSF · LE.A.2

Decision Flowchart: Classify, Then Construct

Step Action
1 Compute consecutive differences (if data allows)
2 If differences constant → linear
3 Compute consecutive ratios
4 If ratios constant → exponential
5 Apply the right construction method
Construct Linear and Exponential Functions · Slide {page}
HSF · LE.A.2

Mixed Example: Classifying and Building from a Table

1 2 3 4
4 8 16 32

Differences: 4, 8, 16 — not constant → not linear.
Ratios: 2, 2, 2 → exponential, , .

Construct Linear and Exponential Functions · Slide {page}
HSF · LE.A.2

Mixed Example: Classifying and Building from a Graph

A graph shows a straight line through and .

  • Visual: straight line → linear
  • (y-intercept readable from graph)
Construct Linear and Exponential Functions · Slide {page}
HSF · LE.A.2

Mixed Example: Two Verbal Descriptions

A. "Population doubles every 10 years from 5,000."

  • Exponential: , →

B. "Tank drains 15 L/min from 300 liters."

  • Linear: , →
Construct Linear and Exponential Functions · Slide {page}
HSF · LE.A.2

Check: Classify and Construct from a Table

;

  1. Compute differences — constant?
  2. Compute ratios — constant?
  3. Classify and construct the function.
  4. Verify one point not used in construction.

Work all four steps before the next slide.

Construct Linear and Exponential Functions · Slide {page}
HSF · LE.A.2

Answer: f(x) = 2 · 3^x

  • Differences: 4, 12, 36 — not constant → not linear
  • Ratios: all equal 3 → exponential,
  • — verify: ✓
Construct Linear and Exponential Functions · Slide {page}
HSF · LE.A.2

Mixed Practice: Six Construction Problems

Tables:

  1. ;
  2. ;

Graphs:

  1. Line through and
  2. Curve through and

Verbal:

  1. "Balance $1,200, earns 6% interest annually"
  2. "Candle 14 in. tall, burns 0.5 in. per hour"
Construct Linear and Exponential Functions · Slide {page}
HSF · LE.A.2

Mixed Practice: Check Your Answers

  1. Linear:
  2. Exponential:
  3. Linear:
  4. Exponential:
  5. Exponential:
  6. Linear:
Construct Linear and Exponential Functions · Slide {page}
HSF · LE.A.2

Complete Summary: Four Representation Types

Input Classify Construct
Points/table Differences or ratios Slope or divide method
Graph Line vs. curve shape Read two integer points
Verbal "amount" vs. "percent" Extract / or /

⚠️ 5% growth → , not
⚠️ first table value if absent
⚠️ Always verify both original points

Construct Linear and Exponential Functions · Slide {page}
HSF · LE.A.2

What's Next: Comparing Growth Rates

Coming up in HSF.LE.A.3:

  • Compare linear and exponential growth over the same interval
  • Discover why exponential always eventually exceeds linear
  • Interpret the long-run implications for real-world models

The functions you constructed here become the examples you'll compare in the next lesson.

Construct Linear and Exponential Functions · Slide {page}