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MathChapter 3: Functions and Linear Equations

Notes

Rate of Change and Slope Notes

Calculate and interpret slope from points, tables, equations, graphs, and real-world units; find intercepts and zeros.

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Slope measures how rapidly a dependent quantity changes for each unit change in an independent quantity. It connects geometric steepness, table patterns, equation coefficients, and real-world rates.

Learning objectives

  • Identify independent and dependent variables.
  • Compute slope consistently from two points.
  • Interpret positive, negative, zero, and undefined slopes.
  • Use tables and units to interpret rate of change.
  • Find intercepts and connect zeros to x-intercepts.

Average rate of change

Slope between two points
\[m=\frac{y_2-y_1}{x_2-x_1}=\frac{\Delta y}{\Delta x}\]

Subtract coordinates in the same order. The denominator cannot be zero for a finite slope.

The independent variable is the input, usually placed on the horizontal axis. The dependent variable responds to it. If cost depends on hours, slope has units dollars per hour, not hours per dollar.

Worked example

Interpret a real-world slope

A reservoir contains \(920\) liters at minute \(4\) and \(740\) liters at minute \(10\). Find and interpret the average rate of change.

  1. Choose a consistent order

    \(\Delta y=740-920=-180\) liters and \(\Delta x=10-4=6\) minutes.

  2. Divide

    \(m=-180/6=-30\).

  3. Interpret

    The negative sign means the volume falls by \(30\) liters per minute on average.

The average rate is \(-30\) liters per minute.

Four slope classifications

Positive, negative, zero, and undefined slope

Read every line from left to right and compare rise with run.

Positive slopePositive slope. The plotted line has slope 0.75 and y-intercept 0.-6-5-4-3-2-1123456-6-5-4-3-2-1123456xyPlotted line
Positive slope
Negative slopeNegative slope. The plotted line has slope -1 and y-intercept 1.-6-5-4-3-2-1123456-6-5-4-3-2-1123456xyPlotted line
Negative slope
Zero slopeZero slope. The plotted line has slope 0 and y-intercept 2.-6-5-4-3-2-1123456-6-5-4-3-2-1123456xyPlotted line
Zero slope
Undefined slopeVertical line x = 2; run is zero, so slope is undefined.-5-4-3-2-112345-5-4-3-2-112345xyVertical line
Undefined slope
Slope classification guide
Line behaviorSlope
Rises left to rightpositive
Falls left to rightnegative
Horizontal; \(\Delta y=0\)\(0\)
Vertical; \(\Delta x=0\)undefined

Slope from a table

A linear table has a constant ratio \(\Delta y/\Delta x\) across every interval. Equal input steps are convenient but not required. If the rate changes, the relation is not linear across the table.

Linear table with constant rate
Time (hours)Distance (miles)
\(1\)\(47\)
\(3\)\(113\)
\(6\)\(212\)

From hours \(1\) to \(3\), the rate is \((113-47)/(3-1)=33\) miles per hour. From \(3\) to \(6\), it is \((212-113)/(6-3)=33\), confirming linearity.

Standard form and intercepts

Standard form
\[Ax+By=C\]

To find the x-intercept, set \(y=0\). To find the y-intercept, set \(x=0\).

Worked example

Find both intercepts

Find the intercepts of \(3x+4y=24\).

  1. x-intercept

    Set \(y=0\): \(3x=24\), so \(x=8\). The point is \((8,0)\).

  2. y-intercept

    Set \(x=0\): \(4y=24\), so \(y=6\). The point is \((0,6)\).

The intercepts are \((8,0)\) and \((0,6)\).
Mini check

Check your understanding

What is the slope through \((-2,5)\) and \((4,-1)\)?

  1. \(-1\)
  2. \(1\)
  3. \(-6\)
  4. \(6\)
Show answer and explanation

Answer: \(-1\)

\(m=(-1-5)/(4-(-2))=-6/6=-1\).

Key takeaways

What to remember

  • Slope is \(\Delta y/\Delta x\) with consistent subtraction order.
  • Attach meaningful output-per-input units.
  • Horizontal slope is zero; vertical slope is undefined.
  • Set the other variable to zero to find an intercept.
  • A function zero is the x-coordinate of an x-intercept.
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Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.