Algebra Ch. 3 Review

Graphing Linear Equation (finding x- & y- intercepts)

Finding The X-Intercept

​For Example: 4x - 2y + 2 = 0

First: Does 4x - 2y + 2 = 0 fit into Ax + By = C?

No. Why?

2 is on the wrong side of the equation. Move 2 to the other side. (Answer on next slide)

Now replace y with 0 to find the x-intercept.

Finding The X-Intercept

For Example: 4x - 2y + 2 = 0

After moving 2 over: 4x - 2y = -2

After replacing y with 0 to find your x-intercept(the x): 4x = -2

.

If you're confused, using the equation 4x - 2y = -2, you replace y with 0, 4x - 2(0) = -2 ---> 4x - 0 = -2 ---> 4x = -2

What is your x?

Multiple Choice

What is your x?

• x = -1/2

• x = -4/2

• x = -2/1

• x = 0

Finding The Y-Intercept

​For Example: 4x - 2y + 2 = 0

We've went over, 4x - 2y + 2 = 0 does not fit into Ax + By = C

4x - 2y + 2 = 0 ---> 4x - 2y = -2

Now replace x with 0 to find the y-intercept.

Finding The Y-Intercept

For Example: 4x - 2y + 2 = 0 ---> 4x - 2y = -2

After replacing x with 0 to find your y-intercept(the y): -2y = -2

If you're confused, using the equation 4x - 2y = -2, if you replace x with 0, 4(0) - 2y = -2 ---> 0 - 2y = -2 ---> -2y = -2

What is your y?

Multiple Choice

what is y?

• y = x

• y = 0

• y = 1

• y = -1

Answer

For Example: 4x - 2y + 2 = 0 ---> 4x - 2y = -2

y = 1

Our equation is now: -2y = -2 (reread past slide or ask me if you're confused)

Now solve for y: -2y = -2

To solve, you divide both sides by -2 ---> y = -2/-2 ---> y = 1

Graphing A Line Parallel To An Axis

How Do You Know If 2 Lines Are Parallel?

​Just Know: Parallel lines have the same slope.

Which two lines are Parallel?

1. y = 6x + 2

2. y = -6x + 2

3. y = 6x - 2

Multiple Select

Which two lines are Parallel?

• y = 6x + 2

• y = -6x + 2

• y = 6x - 2

How Do You Know If 2 Lines Are Parallel?

​Just Know: Parallel lines have the same slope.

Which two lines are Parallel?

1. & 3.

1. y = 6x + 2

2. -----------

3. y = 6x - 2

How Do You Know If 2 Lines Are Parallel? #2

Which two lines are Parallel?

1. y = -1/3x + 1

2. y = -2/6x + 2

3. y = 1/3x +1

Multiple Select

Which two lines are Parallel?

• y = -1/3x + 1

• y = -2/6x + 2

• y = 1/3x +1

How Do You Know If 2 Lines Are Parallel? #2

Which two lines are Parallel?

1. & 2.

1. y = -1/3x + 1

2. y = -2/6x + 2

3. ---------------

   If you're confused on why, simplify -2/6. -2/6 = -1/3

Direct

Variation

Some text here about the topic of discussion

What Is Direct Variation?

Direct Variation(how I interpret it) - How one variable changes in terms of the other.

When they say x in terms of y: x changes as y changes(x depends on y)

When they say y in terms of x: y changes as x changes(y depends on x)

.

To have a direct variation, the line needs to have a constant rate of change or a slope. <--- important

What Is Direct Variation?

x = 2 and y = 4

.

What is x in terms of y? (x/y)

What is y in terms of x? (y/x)

What Is Direct Variation?

x = 2 and y = 4

.

What is x in terms of y? (x/y) : 2/4 or 1/2 (she'll take off points if your answer isn't simplified btw)

What is y in terms of x? (y/x) : 4/2 = 2

Multiple Choice

What is y in terms of x?

y = 6, x = 8

Simplify.

• 3/4

• 6/8

• 4/3

• 8/6

Finding Slope

​Find the slope using these two points.

(1, 3) , (5, 15)

Some text here about the topic of discussion

Multiple Choice

What is x in terms of y?

y = 2.4, x = 2

Simplify.

• 0

• 2/2.4

• 1/1.2

• 1/2

Making Sense Of Direct Variation?

If I said cost(C) varied in terms of the packages sold (P), how would you write that out?

If I said the depth of the water(D) varied in terms of the time (T), how would you write that?

Making Sense Of Direct Variation?

If I said cost(C) varied in terms of the packages sold (P), how would you write that out?

Answer: C/P

If I said the depth of the water(D) varied in terms of the time (T), how would you write that?

Anwer: D/T

Making Sense Of Direct Variation?

The cost(C) varies directly with the number of pencils sold(P).

Find the cost if 12 pencils if 4 pencils cost $6.00.

So first, set up your equation to find the unit rate.

We don't know the cost of 12 pencils but we know the cost of 4.

Using C varies with P, you should have your equation: C/P

With that, substitute each variable with the given info.

Making Sense Of Direct Variation?

The cost(C) varies directly with the number of pencils sold(P).

Find the cost if 12 pencils if 4 pencils cost $6.00.

.

Unit Rate: $1.50 for each pencil

Cost for 12 pencils: $18

.

To find Unit Rate: C/P ---> $6.00/4 = $1.5

Total Cost for 12 pencils: $1.5(12) = $18

Multiple Choice

The cost(C) varies directly with the number of packages(P) sold.

Find the cost of 8 packages if 3 packages cost $16.50

• $12

• $5.50

• $30

• $44

Slopes Of Lines

Multiple Choice

Find the slope using these two points.

(1, 3) , (5, 15)

• m = 4

• m = 3

• m = 0

• m = 2

Finding Slope

(1, 3) , (5, 15)

(1, 3) , (5, 15)

(x1, y1), (x2, y2)

.

Solve: y2-y1/x2-x1

.

15-3/5-1 = 12/4 = 3 m=3

Multiple Choice

What is the slope?

• x = 2/3

• m = 2/3

• y = 3/2

• m = - 2/3

First, find two points on the graph: (3, -2) , (0, -4)

^both points on graph

Using the formula:

y2-y1/x2-x1 ___ substitute

-4 -(-2) / 0 - 3

-4 + 2 / -3

-2/-3 = 2/3

m = 2/3

Multiple Choice

What is the slope?

• y = 3

• x = 3

• y = -3

• x = -3

Horizontal & Vertical Lines

When the line is parallel to the x axis, its y = _

When the line is parallel to the y axis, its x = _

.

In simpler terms, if the line touches the y axis, its y = _

If the line touches the x axis, its x = _

Ik some of you guys struggle w this so

Finding Perpendicular Lines

How Do You Know If 2 Lines Are Perpendicular

Just Know: Perpendicular lines have a slope that's a reciprocal of the other line and the slope has the opposite sign.

Which line is perpendicular to y = 2x - 4

1. y = 1/2x - 4

2. y = -1/2x - 3

3. y = -2/1x - 4

Multiple Choice

Which line is perpendicular to y = 2x - 4?

• y = 1/2x - 4

• y = -1/2x - 3

• y = -2/1x - 4

Answer

Which line is perpendicular to y = 2x - 4

Answer: y = -1/2x - 3

.

The slope in y = 2x - 4 is 2. The reciprocal of 2 is 1/2. And since 2 is positive, the slope of the perpendicular line has to be negative.

m = -1/2

Finding Slope

& y- intercept

Slope & y- intercept

Find the slope and y- intercept of: 3x - 8

The slope is the number right before x. 3 is the slope, not 3x.

The y- intercept is -8.

.

There are other equations where you need to solve the equation first before finding the slope and y- intercept.

Some text here about the topic of discussion

Point Slope Form

What is Point Slope Form?

Point Slope Form: y-y1 = m(x-x1)

If you have problems with this topic, private dm me your questions.

It's hard to explain this topic on slides.

Some text here about the topic of discussion

Finding The Slope-Intercept Form Given Two Points

Finding The Slope-Intercept Form Given Two Points

Given: (1, 4) , (4, 10)

First, find the slope using the points using y2-y1/x2-x1

10-4/4-1 ---> 6/3 = 2 ---- m = 2 (so far we have: y = 2x + ??)

.

4 = 2(1) + ?? <--- this is substituting (1, 4) into y = 2x + ??

So, 4 = 2 + ??, the (??) is the y- intercept, so the y- intercept is 2 as well.

Finding The Slope-Intercept Form Given Two Points

Given: (1, 4) , (4, 10)

Now, substitute (4, 10) into y = 2x + 2

10 = 2(4) + 2 ---> 10 = 8 + 2 (TRUE)

If you're still confused, I made a second example

Finding The Slope-Intercept Form Given Two Points Pt. 2

Given: (2, 10) , (5, 19)

First, find the slope using the points using y2-y1/x2-x1

19-10/5-2 ---> 9/3 = 3 ---- m = 3 (so far we have: y = 3x + ??)

.

10 = 3(2) + ?? <--- this is substituting (2, 10) into y = 3x + ??

So, 10 = 6 + ??, the (??) is the y- intercept, so the y- intercept is 4.

Finding The Slope-Intercept Form Given Two Points Pt. 2

Given: (2, 10) , (5, 19)

Now, substitute (5, 19) into y = 3x + 4

19 = 3(5) + 4 ---> 19 = 15 + 4 (TRUE)

.

Try the next problem.

Multiple Choice

Given the points: (4, 3) , (2, 2)

What is the slope?

y2-y1/x2-x1

• m = 1/2

• m = -1/2

• m = 2/1

• m = -2/1

Multiple Choice

Given the points: (4, 3) , (2, 2)

Slope: 1/2

Substitute to find the y- intercept.

• 0

• -1

• 1

• 1/2

Functions

Functions

How do you determine if a line is a function?

A function:

• passes the vertical line test

• one unique input for every output

• does not have overlapping points in terms of x : (2, 3) , (2, 4)

• x cannot have a number 2x in a row but y can

Algebra Of Functions

Algebra Of Functions

Addition: f(x) + g(x)

Subtraction: f(x) - g(x)

Multiplication: f(x) * g(x)

Division: f(x)/g(x)

set up your equations f(x) + g(x) however the instructions tell you to.

g(x) can be first, they might add in h(x), m(x), but they all represent equations. Just set up the equations matching the variables in the order they present to you in the instructions

Some text here about the topic of discussion

Multiple Choice

f(x) = 2x - 1

g(x) = 3x + 2

What is the sum of f(x) and g(x), h(x)?

Hint: (2x - 1) + (3x + 2)

• x + 1

• 5x + 1

• x + 3

• 5x + 3

Multiple Choice

f(x) = 2x - 1

g(x) = 3x + 2

What is the difference of f(x) and g(x), h(x)?

Hint: (2x - 1) - (3x + 2)

• 5x + 3

• -x + 3

• -x + 1

• 5x + 1

Multiple Choice

f(x) = 2x - 1

g(x) = 3

What is the product of f(x) and g(x), h(x)?

Hint: (2x - 1)(3)

• 6x - 3

• 6x + 3

• 2x - 3

• 2x + 3

Multiple Choice

f(x) = 2x - 4

g(x) = x - 2

What is the quotient of f(x) and g(x), h(x)?

Hint: (2x - 4)/(x - 2)

• x - 2

• 2x - 2

• 2x + 2

• x + 2

Inverse Functions

Steps To Find Inverse Functions

Know: A line has an inverse if it passes the horizontal line test(not the vertical)

Ex: Find the Inverse of y = 2x - 4

.

First, switch the places of your y and x. ---> x = 2y - 4

Now solve for y.

Add 4 to both sides. ---> x + 4 = 2y

Lastly, divide 2 from both sides. ---> (x+4)/2 = y ---> 1/2x + 2 (simplified)

Multiple Choice

Find the inverse function of y = 11x + 22

(x - 22)/11

Simplify.

• y = 1/11x - 2

• No inverse

• y = 1/11x - 22/11

• y = 22/11

Multiple Choice

What is the inverse function of y = 4x - 2?

Simplify.

• y = 1/4x + 2/4

• No inverse

• y = 1/4x + 1/2

• y = x + 1/2

Poll

How confident are you for the upcoming test? 1-10

• 1-3

• 4-6

• 7-8

• 9-10

If You're Still Struggling, Message Me What U Don't Understand