5.9.1 Using the x- intercept

x-intercept as an over arching idea of mathematics Since the study of algebraic mathematics is based on the two variable system we see that mostly what we are interested in studying is the behavior of the dependent(y) variable Since the study of algebraic mathematics is based on the two variable system we see that mostly what we are interested in studying is the behavior of the dependent(y) variable The x-intercept describes the behavior of the dependent(y) variable precisely when the value of the dependent variable is 0. The x-intercept describes the behavior of the dependent(y) variable precisely when the value of the dependent variable is 0. Very helpful for word problems Very helpful for word problems You will see this a lot in mathematics You will see this a lot in mathematics

The one variable system It happens that the two variable system is dependent on the one variable system It happens that the two variable system is dependent on the one variable system Also, for the x-intercept the y = 0 so that the x-intercept, which is a point on a graph, will be of the form (x, 0) Also, for the x-intercept the y = 0 so that the x-intercept, which is a point on a graph, will be of the form (x, 0) Thus for y = mx + b, the slope-intercept form Thus for y = mx + b, the slope-intercept form We see that the x-intercept is found by: We see that the x-intercept is found by: 0 = mx + b and x = 0 = mx + b and x =

Finding the related function Given an equation of one variable, find the related two variable equation in slope- intercept form (problems 1-3) Given an equation of one variable, find the related two variable equation in slope- intercept form (problems 1-3) 10 = 6x – 5 10 = 6x – 5 Find zero on one side of the equation Find zero on one side of the equation 10 = 6x – 5 10 = 6x – = 6x = 6x -15 y = 6x – 15  answer y = 6x – 15  answer

Find the related graph Given an equation of one variable find its related graph. (problems 4-6) Given an equation of one variable find its related graph. (problems 4-6) Solve for x Solve for x x x 5 5 x x 5

Now we know that x = 35 So the related graph must include the point (35, 0), the x-intercept So the related graph must include the point (35, 0), the x-intercept

Solving each equation Solving each equation using a graph. (problems 7-18) Solving each equation using a graph. (problems 7-18) Don’t worry about using a graph, just use algebra Don’t worry about using a graph, just use algebra -3x + 4 = 13 -3x + 4 = x = 9 -3x = x = -3

Proofs and counter examples Writing a proof: Writing a proof: To write a proof you must To write a proof you must state what is given state what is given Show that the conclusion is logical based on what is given (more difficult) Show that the conclusion is logical based on what is given (more difficult) Writing a counter example: Writing a counter example: Show that one specific case does not follow the rules (easier) Show that one specific case does not follow the rules (easier)

Proofs and Counter-examples (problems 27-29) All horizontal lines have an x-intercept All horizontal lines have an x-intercept Given infromation: Given infromation: Since a horizontal line is of the form Since a horizontal line is of the form y = mx + b where m = 0 y = mx + b where m = 0 y = 0x + b y = 0x + b y = 0 + b y = 0 + b y = b y = b Given: x-intercept has y = 0 then Given: x-intercept has y = 0 then 0 = b 0 = b So then a horizontal line has x-intercepts when b = 0 and there for the statement is false, only in the specific case where b = 0 does a horizontal line have an x-intercept So then a horizontal line has x-intercepts when b = 0 and there for the statement is false, only in the specific case where b = 0 does a horizontal line have an x-intercept There for y = mx + b where m = 0 and b ≠ 0 is a counter example There for y = mx + b where m = 0 and b ≠ 0 is a counter example Then y = 3 is a counter example Then y = 3 is a counter example

Homework p all, 13, 14, 27-29all, 34 p all, 13, 14, 27-29all, 34

Chapter 5 review (due May 4?) p all p all