slope problems © 2002 by shawna haider. slope slope the slope of the line passing through the slope...

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Slope Problems © 2002 by Shawna Haider

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Page 1: Slope Problems © 2002 by Shawna Haider. SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is

Slope Problem

s

© 2002 by Shawna Haider

Page 2: Slope Problems © 2002 by Shawna Haider. SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is

SLOPESlope The slope of the line passing through and is given by

1 1( , )x y 2 2( , )x y

2 1 1 2

2 1 1 2

rise verticalm

run horizontaly y y y

x x x x

Page 3: Slope Problems © 2002 by Shawna Haider. SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is

Slope Problem Examples

Determine a value for x such that the line through the points has the given slope.

Let's use the slope formula and plug in what we know.

(x1,y1) (x2,y2)

4

30,5,2, mx

12

12

xx

yym

x

5

)2(0

4

3x

5

2

4

3

You can cross-multiply to find a fraction-free equation for x to solve.

2453 x

Page 4: Slope Problems © 2002 by Shawna Haider. SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is

Example when you have a point and the slope

A point on a line and the slope of the line are given. Find two additional points on the line.

25,1 m

-1 0

Remember that slope is the change in y over the change in x. The slope is 2 which can be made into the fraction

1

2

(0,-3)

So this point is on the line also. You can see that this point is changing (adding) 2 to the y value of the given point and changing (adding) 1 to the x value.

+2+1To find another point on the line, repeat this process with your new point

(0,-3)+1 +2(1,-1) (-1,5)

Page 5: Slope Problems © 2002 by Shawna Haider. SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is

Determine the slope and the y-intercept of the line given by

y mx b

12

3y x

12

3y x

Slope-intercept form

slope

y intercept

m = -1/3 SLOPE

b = 2 y - intercept

Page 6: Slope Problems © 2002 by Shawna Haider. SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is

y in

terc

ept

slop

e

Example of given an equation, find the slope and y intercept

Find the slope and y intercept of the given equation and graph it.

3 1y x

31

1y x

bmxy

Now plot the y

intercept

From the y intercept,

count the slope

Change in y

Change in x

Now that you have 2 points you can draw

the line

Page 7: Slope Problems © 2002 by Shawna Haider. SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is

y in

terc

ept

slop

e

Example of given an equation, find the slope and y intercept

Find the slope and y intercept of the given equation and graph it.

0443 yxFirst let's get this in slope-intercept form by solving for y. -3x +4 -3x +4

434 xy-4 -4

14

3 xy

bmxy

Now plot the y

intercept

From the y intercept,

count the slope

Change in y

Change in x

Now that you have 2 points you can draw

the line

Page 8: Slope Problems © 2002 by Shawna Haider. SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is

These will be linear models. When you read the problem look for two different variables that can be paired together in ordered pairs.

Example: The total sales for a new sportswear store were $150,000 for the third year and $250,000 for the fifth year. Find a linear model to represent the data. Estimate the total sales for the sixth year.

Total sales depend on which year so let’s make ordered pairs (x, y) with x being the year and y being the total sales for that year.

(3, 150,000) (5, 250,000)

We now have two points and can determine a line that contains these the points (on the next screen)

Page 9: Slope Problems © 2002 by Shawna Haider. SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is

We’ll want to use the point-slope equation

(3, 150,000) (5, 250,000)

11 xxmyy First we need the slope

000,502

100000

35

150000250000

12

12

xx

yym

Now we have all the pieces we need so we can plug them in

350000150000 xy

xy 5000015000050000150000 xy

We now have an equation to estimate sales in any given year. To estimate sales in the sixth year plug a 6 in for x.

000,300$650000 y

Page 10: Slope Problems © 2002 by Shawna Haider. SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is

bmxy Slope-Intercept Form

• Useful for graphing since m is the slope and b is the y-intercept 11 xxmyy Point-Slope Form

• Use this form when you know a point on the line and the slope

• Also can use this version if you have two points on the line because you can first find the slope using the slope formula and then use one of the points and the slope in this equation.

0 cbyax General Form

• Commonly used to write linear equation problems or express answers

Page 11: Slope Problems © 2002 by Shawna Haider. SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is

Graphing Horizontal Lines

2 1

2 1 2 1

00

y ym

x x x x

Page 12: Slope Problems © 2002 by Shawna Haider. SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is

Horizontal Lines

The slope of a horizontal line 0The graph of any function of the form f(x) = b or y = b is a horizontal line that crosses

the y-axis at (0, b).

Page 13: Slope Problems © 2002 by Shawna Haider. SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is

Vertical Lines

2 1 2 1

2 2 0

y y y ym undefined

x x

Page 14: Slope Problems © 2002 by Shawna Haider. SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is

Vertical Lines

The slope of a vertical line is undefinedThe graph of any function of the form x =a is a vertical line that crosses the x-axis at

(b, 0).

Page 15: Slope Problems © 2002 by Shawna Haider. SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is

Example of how to find x and y intercepts to graph a line

The x-intercept is where a line crosses the x axis

(6,0)

(-1,0)

(2,0)

What is the common thing you notice about the x-intercepts of these lines?

The y value of the point where they cross the axis will always be 0

To find the x-intercept when we have an equation then, we will want the y value to be zero.

Page 16: Slope Problems © 2002 by Shawna Haider. SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is

Now let's see how to find the y-intercept

The y-intercept is where a line crosses the y axis

(0,4)

(0,1)

(0,5)

What is the common thing you notice about the y-intercepts of these lines?

The x value of the point where they cross the axis will always be 0

To find the y-intercept when we have an equation then, we will want the x value to be zero.

Page 17: Slope Problems © 2002 by Shawna Haider. SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is

Let's look at the equation 2x – 3y = 12

Find the x-intercept. We'll do this by plugging 0 in for y

2x – 3(0) = 12 Now solve for x.

2x = 12

2 2x = 6 So the place where this line

crosses the x axis is (6, 0)

Page 18: Slope Problems © 2002 by Shawna Haider. SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is

2x – 3y = 12

Find the y-intercept. We'll do this by plugging 0 in for x

2(0) – 3y = 12 Now solve for y.

-3y = 12

-3 -3

y = - 4 So the place where this line crosses the y axis is (0, -4)

We now have enough information to graph the line by joining up

these points

(6,0)

(0,- 4)

Page 19: Slope Problems © 2002 by Shawna Haider. SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is

Recognizing Linear Equations

In every equation of the form Ax + By = C linear? To find out, suppose that A and B are nonzero and solve for y:

Ax By C

By Ax C

A Cy x

B B

A, B, and C are constants

Adding –Ax to both sides

Dividing both sides by B

Page 20: Slope Problems © 2002 by Shawna Haider. SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is

Since the last equation is a slope-intercept equation, we see that Ax +By = C is a linear equation when A≠0 and B≠0.But what if A or B (but not both) is 0, then By = C and y = C/B. If B is 0, then Ax = C and x = C/A. In the first case, the graph is a horizontal line; in the second case, the line is vertical. In either case, Ax + By = C is a linear equation when A or B (but not both) is 0.

Standard Form of a Linear Equation

Any equation of the form Ax + By = C, where A, B, and C are real numbers and A and B are not both 0, is a linear equation in standard form and has a graph that is a straight line.

Page 21: Slope Problems © 2002 by Shawna Haider. SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is

Point-Slope Form

1

1

1

1

1

1 1

1

(x x ) ( )

( )

y ym

x x

y ym

x x

y

x

x

x

y m x

Multiply both sides by 1( )x x

Point-Slope Form

Any equation of the form is said to be written in point-slope form and has a graph that is a straight line.

The slope of the line is m

The line passes through .

1 1( )y y m x x

1 1( , )x y

Page 22: Slope Problems © 2002 by Shawna Haider. SLOPE Slope The slope of the line passing through The slope of the line passing through and is given by and is

Remember parallel lines have the same slopes so if you need the slope of a line parallel to a given line, simply find the slope of the given line and the slope you want for a parallel line will be the same.

Perpendicular lines have negative reciprocal slopes so if you need the slope of a line perpendicular to a given line, simply find the slope of the given line, take its reciprocal (flip it over) and make it negative.

3

1m

3

1m

3

1m

31

3m