Let's Learn Difference Quotients - Portland Community College

Home » How To Find Difference Quotient » Let's Learn Difference Quotients - Portland Community College

Maybe your like

Skip to main content\(\require{\require{color,soul}}\newcommand{\abs}[1]{\left\lvert#1\right\rvert} \newcommand{\point}[2]{\left(#1,#2\right)} \newcommand{\highlight}[1]{{\color{blue}{{#1}}}} \newcommand{\highlightr}[1]{{\color{red}{{#1}}}} \newcommand{\highlightg}[1]{{\color{green}{{#1}}}} \newcommand{\highlightp}[1]{{\color{purple}{{#1}}}} \newcommand{\highlightb}[1]{{\color{brown}{{#1}}}} \newcommand{\highlighty}[1]{{\color{gray}{{#1}}}} \newcommand{\lowlight}[1]{{\color{lightgray}{#1}}} \newcommand{\attention}[1]{\mathord{\overset{\downarrow}{#1}}} \newcommand{\substitute}[1]{{\color{blue}{{#1}}}} \newcommand{\addright}[1]{{\color{blue}{{{}+#1}}}} \newcommand{\addleft}[1]{{\color{blue}{{#1+{}}}}} \newcommand{\subtractright}[1]{{\color{blue}{{{}-#1}}}} \newcommand{\multiplyright}[2][\cdot]{{\color{blue}{{{}#1#2}}}} \newcommand{\multiplyleft}[2][\cdot]{{\color{blue}{{#2#1{}}}}} \newcommand{\divideunder}[2]{\frac{#1}{{\color{blue}{{#2}}}}} \newcommand{\divideright}[1]{{\color{blue}{{{}\div#1}}}} \newcommand{\apple}{\text{🍎}} \newcommand{\banana}{\text{🍌}} \newcommand{\pear}{\text{🍐}} \newcommand{\cat}{\text{🐱}} \newcommand{\dog}{\text{🐶}} \newcommand{\uvec}[1]{\boldsymbol{\hat{\textbf{#1}}}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)
  • Front Matter
    • Colophon
    • Acknowledgements
    • About This Site
  • 1 Material Sorted by PCC Course Numbers
    • Topics Covered in MTH 60
    • Topics Covered in MTH 65
    • Topics Covered in MTH 95
    • Topics Covered in MTH 111
    • Topics Covered in MTH 112
  • 2 Absolute Value
    • Videos Created by Emily Nelson
    • Basic Absolute Value Equations
    • Basic Absolute Inequalities
    • Isolating the Absolute Value Expression
    • Additional Practice Solving Absolute Value Equations and Inequalities
  • 3 Exponents and Scientific Notation
    • Videos Created by Ann Cary and Scot Leavitt
    • Properties of Exponents
    • Negative Exponents
    • Summary of the Rules of Exponents
    • Scientific Notation
    • Additional Practice with Exponents and Scientific Notation
  • 4 Factoring
    • Videos Created by Ann Cary and Scot Leavitt
    • Factoring the Greatest Common Factor
    • Factoring Trinomials with Leading Coefficients of 1
    • Factoring by Grouping
    • Factoring Trinomials with Leading Coefficients not 1
    • Factoring Binomials
    • Factoring: A General Strategy
    • Additional Practice Factoring
  • 5 Functions
    • Videos Created by Emily Nelson
    • Videos Created by Ann Cary
    • Introduction to Function Notation
    • Domain and Range of Functions
    • Advanced Function Notation
    • Graphical Transformations
    • Even Functions and Odd Functions
    • Difference Quotients
    • Function Composition
    • Inverse Functions
    • Using Function Graphs to Determine Solution Sets to Inequalities
    • Additional Practice Related to Functions
  • 6 Imaginary and Complex Numbers
    • Videos Created by Emily Nelson
    • Imaginary Numbers
    • Complex Numbers
    • Additional Practice Related to Imaginary and Complex Numbers
  • 7 Lines
    • Videos Created by Ann Cary and Scot Leavitt
    • Intercepts and the Standard Form of the Equation of a Line
    • Slope of a Line
    • The Slope-Intercept Form of the Equation of a Line
    • The Point-Slope Form of the Equation of a Line
    • Parallel Lines, Perpendicular Lines, Horizontal and Vertical Lines
    • Additional Practice Related to Lines
  • 8 Linear Equations and Linear Inequalities in One Variable
    • Introduction to Linear Equations in One Variable
    • Linear Equations in One Variable - Part II
    • Literal Equations
    • Linear Inequalities in One Variable
  • 9 Logarithmic and Exponential Functions
    • Videos Created by Ann Cary
    • Exponential Functions and Their Graphs
    • Logarithmic Functions and Their Graphs
    • Logarithms as Exponents
    • Derivations of the Properties of Logarithms
    • Applying the Properties of Logarithms
    • Logarithmic Equations
    • Exponential Equations
    • The Number \(e\)
    • Exponential Growth and Decay
    • Additional Practice Related to Logarithms and Exponential Equations
  • 10 Parabolas
    • Videos Created by Ann Cary and Scot Leavitt
    • Videos Created by Emily Nelson
    • Standard Form
    • Vertex Form
    • Completing the Square
    • Additional Practice Related to Parabolas
  • 11 Polynomials
    • Videos Created by Ann Cary and Scot Leavitt
    • Polynomial Terminology
    • Addition and Subtraction of Polynomials
    • Multiplication of Polynomials
    • Division of Polynomials by Monomials
    • Additional Practice Related to Polynomials
  • 12 Quadratic Equations
    • Videos Created by Ann Cary and Scot Leavitt
    • Videos Created by Emily Nelson
    • Solving Quadratic Equations Using the Zero-Product Property
    • Solving Quadratic Equations by the Square Root Method
    • Solving Quadratic Equations by Completing the Square
    • Derivation of the Quadratic Formula
    • Applying the Quadratic Formula
    • Complex Number Solutions to Quadratic Equations
    • The Discriminant
    • Applications of Quadratic Equations
    • Additional Practice with Quadratic Equations
  • 13 Radicals and Rational Exponents
    • Videos Created by Ann Cary and Scot Leavitt
    • Videos Created by Emily Nelson
    • Simplifying Square Root Expressions
    • Rationalizing Monomial Square Root Denominators
    • Adding and Subtracting Square Root Expressions
    • Rationalizing Binomial-Denominators that include Square Root expressions
    • Radicals with Indices Greater than Two
    • Rational Exponents
    • Equations Containing Radical Expressions
    • Additional Practice with Radicals and Rational Exponents
  • 14 Rational Expressions, Equations, Functions, and Graphs
    • Videos Created by Emily Nelson
    • Videos Created by Ann Cary
    • Domains of Rational Functions
    • Simplifying Rational Expressions
    • Multiplication and Division of Rational Expressions
    • Addition and Subtraction of Rational Expressions
    • Complex Fractions
    • Rational Equations
    • Rational Functions and Their Graphs
    • Applications of Rational Equations
    • Additional Practice Related to Rational Expressions and Rational Equations
  • 15 Systems of Linear Equations
    • Videos Created by Ann Cary and Scot Leavitt
    • Graphing Systems of Linear Equations
    • Solving Systems Using the Substitution Method
    • Solving Systems Using the Elimination Method
    • Applications of Systems of Linear Equations
    • Additional Practice Related to Systems of Linear Equations
  • 16 Trigonometry
    • The Unit Circle/Angles in Standard Position/Arc Length
    • The Six Basic Trigonometric Functions
    • Special Angles/Reference Angles
    • The Graphs of the Sine and Cosine Functions
    • The Graphs of the Tangent and Cotangent Functions
    • The Graphs of the Secant and Cosecant Functions
    • The Fundamental Trigonometric Identities
    • Multi-Angle Identities
    • Additional Trigonometric Identities
    • Trigonometric Equations
    • The Inverse Trigonometric Functions
    • Right Triangle Trigonometry
    • The Law of Sines and The Law of Cosines
    • Polar Coordinates
    • Complex Numbers/de Moivre's Theorem/Euler's Formula
    • Vectors in 2-Dimensions
  • 17 GeoGebra Applets
    • GeoGebra Applets Master List
    • Determine the Intercepts of a Line Stated in Standard Form
    • Graph a Line Given in Standard Form
    • Create a Line with a Given Slope
    • Determine the Slope of a Line Given Two Points on the Line
    • Graph a Line Given in Slope-Intercept Format
    • Determine the Slope and \(y\)-Intercept of a Line Whose Equation is Stated in Standard Form.
    • Determine the Equation of a Line Given Two Points on the Line
    • Solve a System of Equations Graphically.
    • Solve a System of Equations by Substitution.
    • Solve a System of Equations by Elimination.
    • Explore an Interest Systems Problem Numerically.
    • Explore an Acid Mixture Systems Problem Numerically.
    • Explore a Distance/Rate/Time Systems Problem Numerically.
    • Positive Integer Exponents
    • The Product Rule of Exponents
    • The Power-to-a-Power Rule of Exponents
    • The Power-to-a-Power and Product Rules of Exponents
    • The Quotient Rule of Exponents
    • The Quotient Rule of Exponents
    • The Quotient Rules of Exponents - Which One to Use
    • Practice Exponent Rules Applied to Non-Negative Integer Exponents
    • What's the Meaning of Negative Exponents?
    • Which Factors Move When There's a Negative Exponent?
    • Simplifying Expressions that Contain Integer Exponents
    • Scientific Notation
    • Scientific Notation
    • Scientific Notation Multiplication
    • FOIL
    • Simplify Square Root Expressions.
    • Rationalizing Monomial Denominators That Contain a Square Root Expression
    • Rationalizing Binomial Denominators That Contain Square Root Expressions
    • Explore the Meaning of Rational Exponents
    • Simplifying Square Roots of Negative Integers
    • Multiplication of Complex Numbers
    • Determine the Vertex of a Parabola Given in Standard Form - Part I
    • Determine the Vertex of a Parabola Given in Standard Form - Part II
    • Determine the Vertex of a Parabola Given in Standard Form - Part III
    • Using a Graph to Solve an Inequality - Part I
    • Using a Graph to Solve an Inequality - Part II
    • Determine the Greatest Common Factor
    • Determine the Greatest Common Factor
    • Show the Process of Factoring a Trinomial With a Leading Coefficient of One
    • Practice Factoring Trinomial With Leading Coefficients of One
    • Factoring by Grouping
    • Factoring Trinomials when the Leading Coefficient is not One - Part I
    • Factoring Trinomials when the Leading Coefficient is not One - Part II
    • Factoring Trinomials when the Leading Coefficient is not One - Part III
    • Factoring Difference of Squares that have One Variable
    • Factoring Difference of Squares that have Two Variables
    • Factoring Sum or Difference of Cubes that Have One Variable
    • Factoring Sum or Difference of Cubes that Have Two Variables
    • Introduction to Function Notation - Part One
    • Introduction to Function Notation - Part Two
    • Advanced Function Notation
    • Addition of Rational Expressions
    • Solving Rational Equations
    • Solving Quadratic Equations after Completing the Square
    • Graphing Parabolas when the Equation is in Vertex Form
    • Graphing Parabolas After Completing the Square when the Leading Coefficient is One
    • Graphing Parabolas After Completing the Square when the Leading Coefficient is Two
    • Graphing Parabolas After Completing the Square when the Leading Coefficient is Two
    • Graphical Transformations - Horizontal Shifts
    • Graphical Transformations - Vertical Shifts
    • Graphical Transformations - Horizontal and Vertical Shifts
    • Graphical Transformations - Axis Reflections
    • Graphical Transformations - Horizontal Stretches and Compressions
    • Graphical Transformations - Vertical Stretches and Compressions
    • Graphical Transformations - Stretches, Compressions, and Reflections
    • Graphical Transformations - \(g(x)=a \cdot f(b(x \pm h)) \pm k\)
    • Graphical Transformations - \(g(x)=a \cdot f(b x \pm h) \pm k\)
    • The Unit Circle and the Graph of the Sine Function
    • Unique Triangle - the Case of Side, Side, Side
Authored in PreTeXtPowered by MathJax

Section 5.8 Difference Quotients

Definition of the Difference Quotient.

The difference quotient is a new expression created from a template and a given function formula. The template for the difference quotient of the function \(y=f(x)\) is

\begin{equation*} \frac{f(x+h)-f(x)}{h}\text{.} \end{equation*}

It will be discussed later why this is an important expression. For now, lets just get familiar the algebra associated with difference quotient.

Example 5.8.1.

Determine the difference quotient for the function \(f(x)=8x-7\text{.}\) Make sure that your expression is fully simplified.

Solution

The difference quotient for \(f(x)=8x-7\) follows.

\begin{align*} \frac{\highlight{f(x+h)}-\highlightr{f(x)}}{x}\amp=\frac{(\highlight{8(x+h)-7})-(\highlightr{8x-7})}{h}\\ \amp=\frac{8x+8h-7-8x+7}{h}\\ \amp=\frac{8h}{h}\\ \amp=8,\,\,h \neq 0 \end{align*}
Example 5.8.2.

Determine the difference quotient for the function \(g(t)=6t-t^2\text{.}\) Make sure that your expression is fully simplified.

Solution

The difference quotient for \(g(t)=6t-t^2\) follows.

\begin{align*} \frac{\highlight{g(t+h)}-\highlightr{g(t)}}{h}\amp=\frac{(\highlight{6(t+h)-(t+h)^2})-(\highlightr{6t-t^2})}{h}\\ \amp=\frac{6t+6h-(t+h)(t+h)-6t+t^2}{h}\\ \amp=\frac{6h-(t^2+2th+h^2)+t^2}{h}\\ \amp=\frac{6h-t^2-2th-h^2+t^2}{h}\\ \amp=\frac{6h-2th-h^2}{h}\\ \amp=\frac{h \cdot (6-2t-h)}{h}\\ \amp=-2t+6-h,\,\,h \neq 0 \end{align*}
Example 5.8.3.

Determine the difference quotient for the function \(m(x)=157\text{.}\) Make sure that your expression is completely simplified.

Solution

The difference quotient for the function \(m(x)=157\) follows.

\begin{align*} \frac{\highlight{m(x+h)}-\highlightr{m(x)}}{h}\amp=\frac{\highlight{157}-\highlightr{157}}{h}\\ \amp=\frac{0}{h}\\ \amp=0,\,\,h \neq 0 \end{align*}
Example 5.8.4.

Determine the difference quotient for the function \(w(x)=\frac{5}{x-3}\text{.}\) Make sure that your expression is fully simplified.

Solution

The difference quotient for \(w(x)=\frac{5}{x-3}\) follows.

\begin{align*} \frac{\highlight{w(x+h)}-\highlightr{w(x)}}{h}\amp=\frac{\highlight{\frac{5}{x+h-3}}-\highlightr{\frac{5}{x-3}}}{h}\\ \amp=\frac{\frac{5}{x+h-3} \cdot \highlightb{\frac{x-3}{x-3}}-\frac{5}{x-3} \cdot \highlightb{\frac{x+h-3}{x+h-3}}}{\frac{h}{1}}\\ \amp=\frac{5(x-3)-5(x+h-3)}{(x-3)(x+h-3)} \cdot \frac{1}{h}\\ \amp=\frac{5x-15-5x-5h+15}{(x-3)(x+h-3)h}\\ \amp=\frac{-5 \cdot h}{(x-3)(x+h-3) \cdot h}\\ \amp=-\frac{5}{(x-3)(x+h-3)},\,\, h \neq 0 \end{align*}
The Graphical Significance of the Difference Quotient.

A secant line for the graph of the function \(y=f(x)\) is a line that connects two points on the curve. If you're thinking "gee, it's pretty easy to be a secant line" you're correct in that thought. For example, consider a function that graphs to a parabola. With one exception, any non-vertical line that intersects the parabola once intersects the parabola a second time as well. So with one exception, any non-vertical line that intersects a parabola is a secant line for the parabola. The only exception is the horizontal line that intersects the vertex.

It turns out that the difference quotient is the slope of a secant line to \(f\text{.}\) Specifically, it is the slope of the line connecting the point \((x,f(x))\) with the point \((x+h,f(x+h))\text{.}\)

To understand this, consider \(y=f(x)\text{.}\) Let \(x_1=x\) and \(x_2=x+h\text{.}\) Let's now apply the classic slope-formula.

\begin{align*} m\amp=\frac{y_2-y_1}{x_2-x_1}\\ \amp=\frac{f(x_2)-f(x_1)}{x_2-x_1}\\ \amp=\frac{f(x+h)-f(x)}{(x+h)-x}\\ \amp=\frac{f(x+h)-f(x)}{h} \end{align*}

Note that this implies that for the two stated points the rise of the secant line is

\begin{equation*} \text{rise}=f(x+h)-f(x) \end{equation*}

and the run of the secant line is

\begin{equation*} \text{run}=h \end{equation*}
Example 5.8.5.

Determine the difference quotient for the function \(f(x)=2x^2-x-7\text{.}\) Then use the simplified difference quotient expression to determine the slope of the secant line for \(f\) between the points where \(x=-3\) and \(x=5\text{.}\)

Solution

We begin by determining the simplified difference quotient for \(f\text{.}\)

\begin{align*} \frac{\highlight{f(x+h)}-\highlightr{f(x)}}{h}\amp=\frac{(\highlight{2(x+h)^2-(x+h)-7})-(\highlightr{2x^2-x-7})}{h}\\ \amp=\frac{2(x+h)(x+h)-x-h-7-2x^2+x+7}{h}\\ \amp=\frac{2(x^2+2xh+h^2)-h-2x^2}{h}\\ \amp=\frac{2x^2+4xh+2h^2-h-2x^2}{h}\\ \amp=\frac{4xh+2h^2-h}{h}\\ \amp=\frac{h(4x+2h-1)}{h}\\ \amp=4x+2h-1,\,\,h \neq 0 \end{align*}

The run from the point where \(x=-3\) to the point where \(x=5\) is \(8\text{.}\) So let's take our difference quotient formula and replace \(x\) with \(-3\) and \(h\) with \(8\) to determine the slope of the indicated secant line.

\begin{align*} m\amp=4x+2h-1\\ \amp=4(-3)+2 \cdot 8-1\\ \amp=3 \end{align*}

Let's confirm the slope using the traditional slope formula.

\begin{align*} m\amp=\frac{f(5)-f(-3)}{5-(-3)}\\ \amp=\frac{(2 \cdot 5^2-5-7)-(2 \cdot (-3)^2-(-3)-7)}{8}\\ \amp=\frac{38-14}{8}\\ \amp=\frac{24}{8}\\ \amp=3\,\,\checkmark \end{align*}

Exercises Exercises

Determine the difference quotient for each of the following functions. Make sure that you completely simplify each expression.

1.

\(f(x)=7x-8\)

Solution

The difference quotient for the function \(f(x)=7x-8\) is derived below.

\begin{align*} \frac{\highlight{f(x+h)}-\highlightr{f(x)}}{h}\amp=\frac{\highlight{(7(x+h)-8)}-\highlightr{(7x-8)}}{h}\\ \amp=\frac{7x+7h-8-7x+8}{h}\\ \amp=\frac{7h}{h}\\ \amp=7,\,\,h \neq 0 \end{align*}
2.

\(g(x)=x^2-3x+2\)

Solution

The difference quotient for the function \(g(x)=x^2-3x+2\) is derived below.

\begin{align*} \frac{\highlight{g(x+h)}-\highlightr{g(x)}}{h}\amp=\frac{\highlight{((x+h)^2-3(x+h)+2)}-\highlightr{(x^2-3x+2)}}{h}\\ \amp=\frac{x^2+2xh+h^2-3x-3h+2-x^2+3x-2}{h}\\ \amp=\frac{2xh-3h+h^2}{h}\\ \amp=\frac{h \cdot (2x-3+h)}{h}\\ \amp=2x-3+h,\,\,h \neq 0 \end{align*}
3.

\(g(t)=5-t^2\)

Solution

The difference quotient for the function \(g(t)=5-t^2\) is derived below.

\begin{align*} \frac{\highlight{g(t+h)}-\highlightr{g(t)}}{h}\amp=\frac{\highlight{(5-(t+h)^2)}-\highlightr{(5-t^2)}}{h}\\ \amp=\frac{5-(t^2+2th+h^2)-5+t^2}{h}\\ \amp=\frac{5-t^2-2th-h^2-5+t^2}{h}\\ \amp=\frac{-2th-h^2}{h}\\ \amp=\frac{h \cdot (-2t-h)}{h}\\ \amp=-2t-h,\,\,h \neq 0 \end{align*}
4.

\(k(x)=12\)

Solution

The difference quotient for the function \(k(x)=12\) is derived below.

\begin{align*} \frac{\highlight{k(x+h)}-\highlightr{k(x)}}{h}\amp=\frac{\highlight{12}-\highlightr{12}}{h}\\ \amp=\frac{0}{h}\\ \amp=0,\,\,h \neq 0 \end{align*}
5.

\(r(t)=\frac{3}{t+4}\)

Solution

The difference quotient for the function \(r(t)=\frac{3}{t+4}\) is derived below.

\begin{align*} \frac{\highlight{r(t+h)}-\highlightr{r(t)}}{h}\amp=\frac{\highlight{\frac{3}{t+h+4}}-\highlightr{\frac{3}{t+4}}}{h}\\ \amp=\frac{\frac{3}{t+h+4} \cdot \highlightg{\frac{t+4}{t+4}}-\frac{3}{t+4} \cdot \highlightb{\frac{t+h+4}{t+h+4}}}{h}\\ \amp=\frac{\frac{3t+12-3t-3h-12}{(t+4)(t+h+4)}}{\frac{h}{1}}\\ \amp=\frac{-3h}{(t+4)(t+h+4)} \cdot \frac{1}{h}\\ \amp=\frac{-3 \cdot h}{(t+4)(t+h+4) \cdot h}\\ \amp=-\frac{3}{(t+4)(t+4+h)},\,\, h \neq 0 \end{align*}
6.

\(y(x)=\frac{1}{7-2x}\)

Solution

The difference quotient for the function \(y(x)=\frac{1}{7-2x}\) is derived below.

\begin{align*} \frac{\highlight{y(x+h)}-\highlightr{y(x)}}{h}\amp=\frac{\highlight{\frac{1}{7-2(x+h)}}-\highlightr{\frac{1}{7-2x}}}{h}\\ \amp=\frac{\frac{1}{7-2x-2h}-\frac{1}{7-2x}}{h}\\ \amp=\frac{\frac{1}{7-2x-2h} \cdot \highlightg{\frac{7-2x}{7-2x}}-\frac{1}{7-2x} \cdot \highlightb{\frac{7-2x-2h}{7-2x-2h}}}{h}\\ \amp=\frac{\frac{7-2x-7+2x+2h}{(7-2x-2h)(7-2x)}}{h}\\ \amp=\frac{\frac{2h}{(7-2x-2h)(7-2x)}}{\frac{h}{1}}\\ \amp=\frac{2h}{(7-2x-2h)(7-2x)} \cdot \frac{1}{h}\\ \amp=\frac{2 \cdot h}{(7-2x-2h)(7-2x) \cdot h}\\ \amp=\frac{2}{(7-2x-2h)(7-2x)},\,h\neq 0 \end{align*}

Interpret a difference quotient value as a slope.

7.

The function \(f(x)=\frac{1}{2}x^2+2x-3\) is shown in Figure 5.8.6 along with the secant line connecting the points \((-2,-5)\) and \((2,3)\text{.}\) Determine the difference quotient of \(f\) and then use that to determine the slope of the indicated secant line.

A graph of the function \(f(x)=\frac{1}{2}x^2+2x-3\text{.}\) A line is also graphed. The line intersects the parabola at the points \((-2,-5)\) and \((2,3)\text{.}\)
Figure 5.8.6. \(f(x)=\frac{1}{2}x^2+2x-3\)

Solution

We begin by deriving the difference quotient for \(f(x)=\frac{1}{2}x^2+2x-3\text{.}\)

\begin{align*} \frac{\highlight{f(x+h)}-\highlightr{f(x)}}{h}\amp=\frac{\highlight{\left(\frac{1}{2}(x+h\right)^2+2(x+h)-3)}-\highlightr{\left(\frac{1}{2}x^2+2x-3\right)}}{h}\\ \amp=\frac{\frac{1}{2}(x^2+2xh+h^2)+2x+2h-3-\frac{1}{2}x^2-2x+3}{h}\\ \amp=\frac{\frac{1}{2}x^2+xh+\frac{1}{2}h^2+2h-\frac{1}{2}x^2}{h}\\ \amp=\frac{xh+\frac{1}{2}h^2+2h}{h}\\ \amp=\frac{h \cdot \left(x+\frac{1}{2}h+2\right)}{h}\\ \amp=x+2+\frac{1}{2}h,\,\,h \neq 0 \end{align*}

We now find the slope of the indicated secant line by letting \(x=-2\) and \(h=4\) (the left-most of the two \(x\)-coordinates and the run between the two indicated points). The calculation follows.

\begin{align*} m\amp=-2+2+\frac{1}{2} \cdot 4\\ \amp=2 \end{align*}

A slope of \(2\) is confirmed by looking at the \(\frac{\text{rise}}{\text{run}}\) between the two points.

login

Tag » How To Find Difference Quotient