Section 7.3 Graphs of Functions

Graphs of Functions In this section we are going to look at named graphs that are functions. Names you will get very acquainted with are Linear Function Constant Equation Absolute Value Function Quadratic Function Rational Function Polynomial function

Linear Function A function where the x value and y value do not have any powers. A straight line graph, with no curves Domain will ALWAYS be (-∞, ∞) Range will ALWAYS be (-∞, ∞) The three forms are Standard Form Ax + By = C Slope – Intercept Form y = mx + b Point – Slope Form y - y₁ = m(x - x₁)

Standard Form Standard Form is used for future algebra problems. Ax + By = C Where A, B, C are all rational numbers, no fractions or decimals. Where A needs to be a positive number. Some problems will be Addition Method CH 8 Matrix Method CH 8

Standard Form Example Write 2y + 3/2 = x in standard form 2y + 3/2 = x

Slope – Intercept Form The Slope – Intercept Form is used for graphing the linear function. Y = mx + b m represents the slope Needs to be written in a fraction form Numerator is the up (+) and down (-) movement Denominator is the right (+) and left (-) movement b represents the y-intercept (0, b)

Steps to Graph a Linear Equation 1. Write the equation is slope intercept form 2. Find the y-intercept, (0, b) 3. Plot the y-intercept (0,b). 4. Find the slope, m Write m as a fraction Numerator is the movement on the y-axis, + up, - down Denominator is the movement on the x-axis, + right, - down 5. Use the slope to create your other points. 6. Connect all the points with a line. 7. Label one axis and put all 6 arrows in

Slope – Intercept form Graph y = (1/2) x + 2 Y intercept = (0, 2) Slope = (1/2) up 1 right 2

Slope – Intercept form Graph y = -3x - 1 Y intercept = (0, -1) Slope = -3 = (-3/1) down 3 right 1 = (3/-1) up 3 left 1

Slope – Intercept form Graph y = -2x + 3 Y intercept = Slope =

Slope – Intercept form Graph -3y + x = - 9 Y intercept = Slope =

Point – Slope form Point – Slope form is used with word problems to find the equation of the line. y - y₁ = m(x - x₁) Point 1 (x ₁, y ₁) will change to values Point 2 (x, y) left alone Slope m will change to a value

Point - Slope Form Example Find the equation of the line if the slope is 3 and it goes through the point (1, 2) y - y₁ = m(x - x₁) Point 1 (x ₁, y ₁) = (1, 2) Point 2 (x, y) Slope m = 3 y- 2 = 3(x-1) y – 2 = 3x – 3 y = 3x -1

Point - Slope Form Example Find the equation of the line if the slope is -2 and it goes through the point (-3, 1) y - y₁ = m(x - x₁) Point 1 (x ₁, y ₁) Point 2 (x, y) Slope m

Non-Linear Functions Constant Equation Absolute Value Function Y = # X = # Absolute Value Function f(x) = |x| Quadratic Function f(x) = x² Rational Function f(x) = 1 / x Polynomial Function High degree power…will see later in the semester

Constant function A vertical or horizontal line through a given number. Vertical Line will have the equation x = # Horizontal Line will have the equation y = #

Vertical Line Graph x = 2 Domain Range

Horizontal Line Graph y = 2 Domain Range

Absolute Value Function F(x) = |x|

Absolute Value Function F(x) = |x| Domain Range

Quadratic Function F(x) = x²

Quadratic Function F(x) = x² Domain Range

Rational Function F(x) = 1 / x

Rational Function F(x) = 1 / x Domain Range

Homework 7, 9, 12, 15, 17, 20, 25, 32, 41, 46, 47, 64, 65