2.7 – Absolute Value Inequalities To solve an absolute value inequality, treat it as an equation. Create two inequalities, where the expression in the absolute value is set to be positive and negative.

2.7 – Absolute Value Inequalities Alternate Method If a is a positive number, then  X  < a is equivalent to  a < x < a.

2.7 – Absolute Value Inequalities Example Solve  x + 4  < 6  6 < x + 4 < 6  6 – 4 < x + 4 – 4 < 6 – 4  10 < x < 2 (  10, 2)

2.7 – Absolute Value Inequalities Solve  x  3  + 6  7  x  3   1  1  x  3  1 2  x  4 [2, 4]  x  3     x   Example

2.7 – Absolute Value Inequalities Solve  8x  3  <  2 No solution An absolute value cannot be less than a negative number, since it can’t be negative. Example

2.7 – Absolute Value Inequalities If a is a positive number, then  X  > a is equivalent to X a. or

2.7 – Absolute Value Inequalities Solve  x  + 1 > 2 Example  x  > x <  1 3x <  113x >  9 (  , )  (  3,  ) or10 + 3x > 1

2.7 – Absolute Value Inequalities Example Solve or LCD: 2

Defn: A relation is a set of ordered pairs. Domain: The values of the 1 st component of the ordered pair. Range: The values of the 2nd component of the ordered pair. 3.2 – Introduction to Functions

xy xy xy State the domain and range of each relation.

The Rectangular Coordinate System Ordered Pair (x, y) (independent variable, dependent variable) (1 st component, 2 nd component) (input, output) (abscissa, ordinate) 3.2 – Introduction to Functions

x y 1 st Quadrant2 nd Quadrant 3 rd Quadrant4 th Quadrant The Rectangular Coordinate System 3.2 – Introduction to Functions

Defn: A function is a relation where every x value has one and only one value of y assigned to it. xy xy xy functionnot a function function State whether or not the following relations could be a function or not. 3.2 – Introduction to Functions

Functions and Equations. xy xy xy function not a function 3.2 – Introduction to Functions State whether or not the following equations are functions or not.

3.2 – Introduction to Functions Vertical Line Test Graphs can be used to determine if a relation is a function. If a vertical line can be drawn so that it intersects a graph of an equation more than once, then the equation is not a function.

x y The Vertical Line Test xy – Introduction to Functions function

x y xy – Introduction to Functions The Vertical Line Test function

x y xy – Introduction to Functions The Vertical Line Test not a function

Find the domain and range of the function graphed to the right. Use interval notation. x y Domain: Domain Range: Range [ – 3, 4] [ – 4, 2] 3.2 – Introduction to Functions Domain and Range from Graphs

Find the domain and range of the function graphed to the right. Use interval notation. x y Domain: Domain Range: Range (– ,  ) [– 2,  ) 3.2 – Introduction to Functions Domain and Range from Graphs

3.2 – Introduction to Functions Function Notation Shorthand for stating that an equation is a function. Defines the independent variable (usually x) and the dependent variable (usually y).

Function notation also defines the value of x that is to be use to calculate the corresponding value of y. 3.2 – Introduction to Functions f(x) = 4x – 1 find f(2). f(2) = 4(2) – 1 f(2) = 8 – 1 f(2) = 7 (2, 7) g(x) = x 2 – 2x find g(–3). g(–3) = (-3) 2 – 2(-3) g(–3) = g(–3) = 15 (–3, 15) find f(3).

Given the graph of the following function, find each function value by inspecting the graph. f(5) = 7 x y f(x)f(x) f(4) = 3 f(  5) = 11 f(  6) = 66 3.2 – Introduction to Functions ● ● ● ●

3.3 – Graphing Linear Functions Identifying Intercepts The graph of y = 4x – 8 is shown below. The intercepts are: (2, 0) and (0, –8). The graph crosses the y-axis at the point (0, –8). Likewise, the graph crosses the x-axis at (2, 0). This point is called the y-intercept. This point is called the x-intercept.

3.3 – Graphing Linear Functions Finding the x and y intercepts To find the x-intercept: let y = 0 or f(x) = 0 and solve for x. To find the y-intercept: let x = 0 and solve for y Given: 4 = x – 3y, find the intercepts x-intercept, let y = 0 4 = x – 3(0) 4 = x x-intercept: (4,0). y-intercept, let x = 0 4 = 0 – 3y 4 = –3y y-intercept: (0,-4/3). –4/3 = y 

3.3 – Graphing Linear Functions Plotting the x and y intercepts x y The graph of 4 = x – 3y is the line drawn through these points. Plot the two intercepts and (4, 0) and. (4, 0) (0, )

3.3 – Graphing Linear Functions Vertical Lines Graph the linear equation x = 3. xy Standard form as x + 0y = 3. No matter what value y is assigned, x is always 3.

3.3 – Graphing Linear Functions Horizontal Lines Graph the linear equation y = 3. Standard form as 0x + y = 3. No matter what value x is assigned, y is always 3. xy

3.4 – The Slope of a Line Slope of a Line The slope m of the line containing the points (x 1, y 1 ) and (x 2, y 2 ) is given by:

3.4 – The Slope of a Line Find the slope of the line through (0, 3 ) and (2, 5). Graph the line.

3.4 – The Slope of a Line Find the slope of the line containing the points (4, –3 ) and (2, 2). Graph the line.

3.4 – The Slope of a Line Only a linear equation in two variables can be written in slope-intercept form, y = mx + b. Slope-Intercept Form M is the slope of the line and b (0, b) is the y-intercept of the line. slopey-intercept is (0, b)

Find the slope and the y-intercept of the line Example Solve the equation for y. 3.4 – The Slope of a Line The slope of the line is 3/2. The y-intercept is (0, 11/2).

Find the slope of the line x = 6. Vertical line Use two points (6, 0) and (6, 3). Example The slope is undefined. 3.4 – The Slope of a Line

Find the slope of the line y = 3. Horizontal line Use two points (0, 3) and (3, 3). Example The slope is zero. 3.4 – The Slope of a Line

Slopes of Vertical and Horizontal Lines The slope of any vertical line is undefined. 3.4 – The Slope of a Line The slope of any horizontal line is 0.

Appearance of Lines with Given Slopes Positive Slope Line goes up to the right x y Lines with positive slopes go upward as x increases. Negative Slope Line goes downward to the right x y Lines with negative slopes go downward as x increases. m > 0 m < – The Slope of a Line

Appearance of Lines with Given Slopes Zero Slope horizontal line x y Undefined Slope vertical line x y 3.4 – The Slope of a Line

Parallel Lines & Perpendicular Lines x y 3.4 – The Slope of a Line Two non-vertical lines are perpendicular if the product of their slopes is –1. x y Two non-vertical lines are parallel if they have the same slope and different y- intercepts.

Parallel Lines & Perpendicular Lines 3.4 – The Slope of a Line

Are the following lines parallel, perpendicular, or neither? Example The lines are perpendicular. Parallel Lines & Perpendicular Lines 3.4 – The Slope of a Line x + 5y = 5–5x + y = –6

y = mx + b has a slope of m and has a y-intercept of (0, b). Slope-Intercept Form 3.5 – Equations of Lines This form is useful for graphing, as the slope and the y-intercept are readily visible.

Example Graph The slope is 1/4. The y-intercept is (0, –3) Plot the y-intercept. Slope = rise over run. Rise 1 unit; run 4 units right The graph runs through the two points. 3.5 – Equations of Lines

Example Write an equation of the line with y-intercept (0, –5) and slope of 2/ – Equations of Lines

The point-slope form allows you to use ANY point, together with the slope, to form the equation of the line. Point-Slope Form m is the slope 3.5 – Equations of Lines (x 1, y 1 ) is a point on the line

Find an equation of a line with slope – 2, containing the point (–11, –12). Write the equation in point-slope form and slope-intercept form. Example Point-Slope Form 3.5 – Equations of Lines y – (–12) = – 2(x – (–11)) y + 12 = –2x – 22 y =  2x – 34 y + 12 = – 2(x + 11) Slope-intercept Form Point-slope Form

Example Point-Slope Form 3.5 – Equations of Lines Find an equation of the line through (–4, 0) and (6, –1). Write the equation using function notation.

Example Point-Slope Form 3.5 – Equations of Lines

Example Point-Slope Form 3.5 – Equations of Lines Write the equation of the line graphed. Write the equation in standard form. Continued. ● ● Identify two points on the line. (2, 5) and(  4, 3) Find the slope.

Example 3.5 – Equations of Lines (2, 5) and(  4, 3)

Example 3.5 – Equations of Lines (2, 5) and(  4, 3) LCD: 3

Example 3.5 – Equations of Lines Find an equation of the horizontal line containing the point (2, 3). ● The equation is y = 3.

Example 3.5 – Equations of Lines Find an equation of the line containing the point (6, -2) with undefined slope. ● What type of line has undefined slope? The equation is x = 6. Horizontal or Vertical