5.9.1 Using the x-5.9.1 Using the x-interceptintercept

x-intercept as an over arching idea x-intercept as an over arching idea of mathematicsof mathematics

Since the study of algebraic mathematics Since the study of algebraic mathematics is based on the two variable system we is based on the two variable system we see that mostly what we are interested in see that mostly what we are interested in studying is the behavior of the studying is the behavior of the dependent(y) variabledependent(y) variable

The x-intercept describes the behavior of The x-intercept describes the behavior of the dependent(y) variable precisely when the dependent(y) variable precisely when the value of the dependent variable is 0.the value of the dependent variable is 0.

Very helpful for word problemsVery helpful for word problems You will see this a lot in mathematicsYou will see this a lot in mathematics

The one variable systemThe one variable system It happens that the two variable system is It happens that the two variable system is

dependent on the one variable systemdependent on the one variable system Also, for the x-intercept the y = 0 so that Also, for the x-intercept the y = 0 so that

the x-intercept, which is a point on a the x-intercept, which is a point on a graph, will be of the form (x , 0)graph, will be of the form (x , 0)

Thus for y = mx + b, the slope-intercept Thus for y = mx + b, the slope-intercept formform

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 = mb

Finding the related functionFinding the related function Given an equation of one variable, find the Given an equation of one variable, find the

related two variable equation in slope-related two variable equation in slope-intercept form (problems 1-3)intercept form (problems 1-3)

10 = 6x – 510 = 6x – 5 Find zero on one side of the equationFind zero on one side of the equation 10 = 6x – 510 = 6x – 5 -10 -10-10 -10 0 = 6x -150 = 6x -15 y = 6x – 15 y = 6x – 15 answer answer

Find the related graphFind the related graph Given an equation of one variable Given an equation of one variable

find its related graph. (problems 4-6)find its related graph. (problems 4-6)

Solve for xSolve for x

+9 +9+9 +9

12953

x

2153

x

12953

x

2153

x 5 x x 55 x x 5

1053 x

35x

Now we know that x = 35Now we know that x = 35 So the related graph must include So the related graph must include

the point (35, 0), the x-interceptthe point (35, 0), the x-intercept

Solving each equationSolving each equation Solving each equation using a graph. Solving each equation using a graph.

(problems 7-18)(problems 7-18) Don’t worry about using a graph, just Don’t worry about using a graph, just

use algebrause algebra -3x + 4 = 13-3x + 4 = 13 -4 -4-4 -4 -3x = 9-3x = 9 -3 -3-3 -3 x = -3x = -3

Proofs and counter examplesProofs and counter examples Writing a proof:Writing a proof: To write a proof you must To write a proof you must state what is givenstate what is given Show that the conclusion is logical Show that the conclusion is logical

based on what is given (more based on what is given (more difficult)difficult)

Writing a counter example:Writing a counter example: Show that one specific case does not Show that one specific case does not

follow the rules (easier)follow the rules (easier)

Proofs and Counter-examples Proofs and Counter-examples (problems 27-29)(problems 27-29)

All horizontal lines have an x-interceptAll horizontal lines have an x-intercept Given infromation:Given infromation: Since a horizontal line is of the formSince a horizontal line is of the form y = mx + b where m = 0y = mx + b where m = 0 y = 0y = 0x + bx + b y = 0 + by = 0 + b y = by = b Given: x-intercept has y = 0 thenGiven: x-intercept has y = 0 then 0 = b0 = b So then a horizontal line has x-intercepts when b = 0 So then a horizontal line has x-intercepts when b = 0

and there for the statement is false, only in the and there for the statement is false, only in the specific case where b = 0 does a horizontal line have specific case where b = 0 does a horizontal line have an x-interceptan x-intercept

There for y = mx + b where m = 0 and b ≠ 0 is a There for y = mx + b where m = 0 and b ≠ 0 is a counter example counter example

Then y = 3 is a counter exampleThen y = 3 is a counter example

HomeworkHomework p. 258 1-10all, 13, 14, 27-29all, 34 p. 258 1-10all, 13, 14, 27-29all, 34

Chapter 5 review (due May 4?)Chapter 5 review (due May 4?) p. 261 1-40 all p. 261 1-40 all