Where we’ve been… Today, we take our first step into Chapter 3: Linear Functions – which is fundamentally at the of Algebra Before we take our first step, let’s go back and see what objectives we accomplished in Chapter 2. In Chapter 2, we… Solved equations by using the 4 basic operations Solved one-step, two-step, and multi-step equations Used formulas to solve real world problems

Now, in Chapter 3, our objective will be to… 1. Identify linear equations 2. Identify x- and y- intercepts 3. Graph linear equations using the x- and y-intercepts or a table of values with 4 domain values

3.1 Graphing Linear Equations Our objective: To identify a linear equation in Standard Form To identify the X-intercept and Y-intercept of a linear equation

Identifying a Linear Equation Ax + By = C or Ax – By = C The exponent of each variable is 1. The variables are added or subtracted. A or B can equal zero. A > 0 (cannot have a negative coefficient) Besides x and y, other commonly used variables are m and n, a and b, and r and s. There are no radicals (sq. root symbols) in the equation. Every linear equation graphs as a line.

Examples of linear equations 2x + 4y =8 6y = 3 – x x = 1 -2a + b = 5 Equation is in Ax + By =C form Rewrite with both variables on left side … x + 6y =3 B =0 … x + 0 y =1 Multiply both sides of the equation by -1 … 2a – b = -5 Multiply both sides of the equation by 3 … 4x –y =-21

Examples of Nonlinear Equations The following equations are NOT in the standard form of Ax + By =C: The exponent is 2 There is a radical in the equation Variables are multiplied Variables are divided (can’t have a variable in the denominator) 4x2 + y = 5 xy + x = 5 s/r + r = 3

x and y -intercepts The x-intercept is the point where a line crosses the x-axis. The general form of the x-intercept is (x, 0). The y-coordinate will always be zero. The y-intercept is the point where a line crosses the y-axis. The general form of the y-intercept is (0, y). The x-coordinate will always be zero.

Finding the x-intercept For the equation 2x + y = 6, we know that y must equal 0. What must x equal? Plug in 0 for y and simplify. 2x + 0 = 6 2x = 6 x = 3 So (3, 0) is the x-intercept of the line.

Finding the y-intercept For the equation 2x + y = 6, we know that x must equal 0. What must y equal? Plug in 0 for x and simplify. 2(0) + y = 6 0 + y = 6 y = 6 So (0, 6) is the y-intercept of the line.

To summarize…. To find the x-intercept, plug in 0 for y. To find the y-intercept, plug in 0 for x.

Find the x and y- intercepts of x = 4y – 5 Plug in x = 0 x = 4y - 5 0 = 4y - 5 5 = 4y = y (0, ) is the y-intercept x-intercept: Plug in y = 0 x = 4y - 5 x = 4(0) - 5 x = 0 - 5 x = -5 (-5, 0) is the x-intercept

Find the x and y-intercepts of g(x) = -3x – 1* Plug in x = 0 g(x) = -3(0) - 1 g(x) = 0 - 1 g(x) = -1 (0, -1) is the x-intercept Plug in y = 0 g(x) = -3x - 1 0 = -3x - 1 1 = -3x = x ( , 0) is the *g(x) is the same as y

Find the x and y-intercepts of 6x - 3y =-18 x-intercept Plug in y = 0 6x - 3y = -18 6x -3(0) = -18 6x - 0 = -18 6x = -18 x = -3 (-3, 0) is the y-intercept Plug in x = 0 6x -3y = -18 6(0) -3y = -18 0 - 3y = -18 -3y = -18 y = 6 (0, 6) is the

Find the x and y-intercepts of x = 3 x-intercept Plug in y = 0. There is no y. Why? x = 3 is a vertical line so x always equals 3. (3, 0) is the x-intercept. y-intercept A vertical line never crosses the y-axis. There is no y-intercept. x y

Find the x and y-intercepts of y = -2 y = -2 is a horizontal line so y always equals -2. (0,-2) is the y-intercept. x-intercept Plug in y = 0. y cannot = 0 because y = -2. y = -2 is a horizontal line so it never crosses the x-axis. There is no x-intercept. x y