Chapter 2 Linear and Quadratic Functions, Precalculus 3e Stitz/Zeager

Author

Chalmeta

2.1 Linear Equations in Two Variables

Slope-intercept form

NoteSlope-intercept form of a line

The simplest mathematical model is the linear equation in two variables. The standard form is (slope-intercept):

\[\boxed{y = mx +b}\]

where \(m\) is the slope and \(b\) is the \(y\)-intercept. To calculate the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) the formula is:

\[m = \frac{y_2 -y_1}{x_2 - x_1}=\frac{\Delta y}{\Delta x} = \frac{\text{rise}}{\text{run}}\]

The slope is the amount of vertical change relative to the horizontal change. Sometimes we think of it as the “change in \(y\)” over “change in \(x\)”.

Positive Slope

Graph showing a line with positive slope on a coordinate plane. The line passes through the origin and rises from left to right at approximately a 45-degree angle. A right triangle is drawn on the line showing the rise (Δy, vertical leg) and run (Δx, horizontal leg), illustrating that slope = rise/run. The positive slope indicates that as x increases, y also increases.

Negative Slope

Graph showing a line with negative slope on a coordinate plane. The line passes through the origin and falls from left to right at approximately a 45-degree downward angle. A right triangle is drawn on the line showing the rise (Δy, vertical leg pointing downward) and run (Δx, horizontal leg), illustrating that slope = rise/run where rise is negative. The negative slope indicates that as x increases, y decreases.

TipExample 2.1.1

Sketch the graphs of the following two functions:

\[y=2x+3 \qquad \qquad y = -\frac{1}{2} x +3\]

Empty coordinate plane with gridlines. The x-axis and y-axis are labeled and extend from negative to positive values, with grid squares marked. This blank grid is provided for students to sketch the linear function.

Empty coordinate plane with gridlines. The x-axis and y-axis are labeled and extend from negative to positive values, with grid squares marked. This blank grid is provided for students to sketch the linear function.

TipExample 2.1.2

Find the slope between the following pairs of points.

  1. (-3, 0) and (4, 4)

  2. (-3, 1) and (4, 1)

  3. (-3, 1) and (-3, 4)

Point-Slope Form

NotePoint-slope Form of a Line

The point-slope form of the line is written as:

\[y - y_1 = m (x - x_1)\]

You always need two things:

  1. a point: \((x_1, y_1)\) AND
  2. a slope \(m\).
TipExample 2.1.3

Write the equation of the line through (-3, 0) and (4, -4). Write the equation in the point slope form and the slope-intercept form.

TipExample 2.1.4

Write the equation of the lines through:

  1. (-3, 1) and (4, 1)

  2. (-3, 1) and (-3, 4)

Parallel and Perpendicular Lines

NoteParallel and perpendicular lines

Parallel lines have the same slope. If \(y = m_1 x + b_1\) is parallel to \(y = m_2 x + b_2\) then \(m_1 = m_2\).

Perpendicular lines have negative reciprocal slopes. If \(y = m_1 x + b_1\) is perpendicular to \(y = m_2 x + b_2\) then \(m_1 = -\frac{1}{m_2}\) and \(m_2 = -\frac{1}{m_1}\).

TipExample 2.1.5

Write the equations of the lines parallel and perpendicular to \(-4x + 2y = 3\) passing through the point (2, 1).

2.2 Absolute Value Functions

NoteAbsolute Value Definition

Definition: The absolute value of a real number \(x\), denoted \(|x|\), is given by:

\[|x| = \begin{cases} -x, & \text{if } x < 0 \\ x, & \text{if } x \geq 0 \end{cases}\]

Absolute Value Properties

  1. Product rule: \(|ab| = |a| |b|\)
  2. Power rule: \(|a^n| = |a|^n\)
  3. Quotient rule: \(\left|\frac{a}{b}\right| = \frac{|a|}{|b|}\)
  4. Equality property 1: \(|x|=0\) if and only if \(x=0\)
  5. Equality property 2: For \(c>0\), \(|x|=c\) if and only if \(x=c\) or \(x = -c\).
  6. Equality property 3: For \(c<0\), \(|x|=c\) has no solution.

An equation with an absolute value is always TWO equations:

\[|x-5| = 4 \qquad \implies \qquad x-5=4 \text{ OR } -(x-5) = 4\]

TipExample 2.2.1

\[-4+2|4x-4| = 10\]

We start by getting the absolute value by itself on one side of the equation.

ALWAYS check your solutions:

TipExample 2.2.2

Sketch a graph of \(f(x) = \frac{1}{2} |x-1|-3\)

Coordinate plane with gridlines, x-axis ranging from approximately -4 to 4 and y-axis ranging from approximately -4 to 5. This empty grid is provided for students to sketch the absolute value function f(x) = ½|x-1|-3, which is a V-shaped graph with vertex at (1, -3), opening upward with a slope of ½ on the right side and -½ on the left side.

2.3 Quadratic Functions

NoteQuadratic Equations

Definition: Let \(a\), \(b\), and \(c\) be real numbers with \(a \neq 0\). The function:

\[f(x) = ax^2 + bx + c\]

is called a quadratic equation.

The graph of a quadratic equation is a parabola. All parabolas are symmetric with respect to the axis of symmetry which passes through the vertex.

TipExample 2.3.1

\[f(x)=x^2+2x+1 \qquad \text{OR} \qquad f(x) = -x^2-2x-1\]

Parabola opening upward on a coordinate plane. The parabola has its vertex below the x-axis and extends upward on both sides. A vertical dashed line passes through the vertex, representing the axis of symmetry. The coefficient a is positive, causing the parabola to open upward in a U-shape.

\(a>0\) graph opens up

Parabola opening downward on a coordinate plane. The parabola has its vertex above the x-axis and extends downward on both sides. A vertical dashed line passes through the vertex, representing the axis of symmetry. The coefficient a is negative, causing the parabola to open downward in an inverted U-shape.

\(a<0\) graph opens down

NoteThe Standard Form of a Parabola

The Standard Form of a Parabola

\[f(x)=a(x-h)^2+k\]

Vertex is located at \((h, k)\)

if \(a>0\) graph opens up

if \(a<0\) graph opens down

Alternate form

If \(f(x)= ax^2+bx+c\) then the:

\[\text{vertex} = \left(\frac{-b}{2a}, ~ f\left(\frac{-b}{2a}\right)\right)\]

To Graph a Parabola

  1. Find the vertex
  2. Find the \(x\)-intercepts
  3. Determine if it opens up or down
  4. Sketch
TipExample 2.3.2

Graph \(f(x) = 2x^2-12x-14\).

Step 1: Vertex:

\(x = \frac{-(-12)}{2(2)} = 3\)

\(y = f(3) = 2(9)-12(3)-14 = -32\)

So the coordinates of the vertex are \((x, y) = (3, -32)\)

Step 2: \(x\)-intercepts:

\[0 = 2x^2-12x-14\]

Step 4: Sketch

Empty coordinate plane with gridlines. The x-axis and y-axis are marked, with the grid extending to show the relevant range for graphing the parabola f(x) = 2x²-12x-14, which has vertex at (3, -32) and opens upward.

TipExample 2.3.3

Graph \(f(x)=(x-6)^2+3\)

Step 4: Sketch

Coordinate plane with gridlines showing a smaller range. The x-axis ranges from approximately -1 to beyond the visible area on the right, and the y-axis shows positive values. This grid is for graphing the parabola f(x)=(x-6)²+3, which has vertex at (6, 3) and opens upward.

TipExample 2.3.4

Find the standard form for a parabola that has (0, 1) as its vertex and passes through the point (1, 0)

TipExample 2.3.5

Convert to standard form \(f(x) = x^2+6x +5\) and graph.

TipExample 2.3.6

Convert to standard form \(f(x) = x^2-2x -8\) and graph.

TipExample 2.3.7

Factor the following:

  1. \(9x\cdot (x-3)+7\cdot (x-3)\)
  1. \(x^3 - 8x^2+15x\)
  1. \(x^2+11x+28\)
  1. \(6t^2+5t -4\)
  1. \(5z^2+23z+12\)
TipExample 2.3.8

A rancher has 260 yards of fence with which to enclose three sides of a rectangular meadow (the fourth side is a river and will not require fencing). Find the dimensions of the meadow with the largest possible area.

TipExample 2.3.9

A person standing close to the edge on top of a 72-foot building throws a ball vertically upward. The quadratic function \(h(t) = -16t^2 +84t+72\) models the ball’s height about the ground, \(h(t)\), in feet, \(t\) seconds after it was thrown.

  1. What is the maximum height of the ball?

  2. How many seconds does it take until the ball hits the ground?

2.4 Solving Inequalities with Absolute Value and Quadratic Functions

Graphing inequalities

Expression Number Line
\(x > 2\) Number line from -4 to 6 for graphing inequality solutions
\(x \geq 2\) Number line from -4 to 6 for graphing inequality solutions
\(x \geq 2\) AND \(x \leq 4\) Number line from -4 to 6 for graphing inequality solutions
Number line from -4 to 6 for graphing inequality solutions
Number line from -4 to 6 for graphing inequality solutions
\(x \leq 2\) OR \(x \geq 4\) Number line from -4 to 6 for graphing inequality solutions
Number line from -4 to 6 for graphing inequality solutions
Number line from -4 to 6 for graphing inequality solutions
\(x \leq 2\) AND \(x \geq 4\) Number line from -4 to 6 for graphing inequality solutions

Interval Notation

Inequality notation Concise notation Interval notation
\(x > 2\) \((2, \infty)\)
\(x \geq 2\) \([2, \infty)\)
\(x \geq 2\) AND \(x \leq 4\) \(2 \leq x \leq 4\) \([2, 4]\)
\(x \leq 2\) OR \(x \geq 4\) \((-\infty, 2] \cup [4, \infty)\)
NoteProperties of Inequalities

Properties of Inequalities

  1. Transitive: \(a < b\) and \(b < c \implies a < c\)
  2. Addition of Constants: If \(a < b\) then \(a+ c < b + c\)
  3. Addition: If \(a < b\) and \(c < d\) then \(a +c < b + d\)
  4. Multiplication by a constant:
    • If \(c > 0\) and \(a < b\) then \(ac < bc\)
    • If \(c < 0\) and \(a < b\) then \(ac > bc\)

NOTE: If you multiply or divide by a negative number you reverse the order of the inequality.

Solving linear inequalities

TipExample 2.4.1

\(10x < 40\)

Number line from -4 to 6 for showing solution to 10x < 40

TipExample 2.4.2

\(-10x < 40\)

Number line from -4 to 6 for showing solution to -10x < 40. Note: dividing by negative reverses the inequality

TipExample 2.4.3

\(4(x+1) \leq 2x+3\)

Number line from -4 to 6 for showing solution to 4(x+1) ≤ 2x+3

TipExample 2.4.4

\(-8 \leq -(3x+5) < 13\)

Number line from -4 to 6 for showing solution to compound inequality -8 ≤ -(3x+5) < 13

Absolute value and inequalities

Absolute value is still two equations

TipExample 2.4.5

\(\left|\frac{x}{2}\right| > 5\)

Number line from -15 to 10 for showing solution to absolute value inequality |x/2| > 5

\[\frac{x}{2} > 5 \qquad \text{OR} \qquad -\frac{x}{2} > 5\]

TipExample 2.4.6

\(|x-7| < 5\)

Number line from 0 to 14 for showing solution to absolute value inequality |x-7| < 5

\[-5 < x-7 < 5\]

Question: What does \(|x-2| < 5\) mean?

Answer: All real numbers within five units of two.

So all real numbers within 5 units of 8 would be written as:

And all real numbers at least 5 units from 8 would be written as:

Solving polynomial inequalities

TipExample 2.4.7

\(x^2 < 5\)

Step 1: Set equation equal to zero and find the zeros. (Factor)

Step 2: Set up a table of signs

Step 3: Find where the table gives negative values and write the solution

Number line from -8 to 3 for showing solution to polynomial inequality x² < 5

TipExample 2.4.8

\(x^2 + 2x -3 \geq 0\)

TipExample 2.4.9

\((x-1)^2(x+2)^3 \geq 0\)