Linear Functions
Algebra I Unit 5
Warm Up for 1/3
Slope
• What are some different ways we can think about finding the slope of
a line?
• Slope is…
Slope
• For each pair of points, find the missing x or y value that lets the
points have the given slope
(7, 4), (3, y); slope =
1
4
(x, 5), (–3, 6); slope = 1
(5, y), (6, 4); slope = 0
(12, 9), (x, –2); slope =
1
−
2
Warm Up for 1/4
• Find the slope of the lines between the given pairs of points
(5,3) (-2, -4)
(7,1) (7, -4)
(3,8) (7,3)
(2,3) (6,5)
Slope Intercept Form
Slope Intercept Form
Warm Up for 1/5
• Graph each slope intercept equation.
𝑦 = −4𝑥 + 2
𝑦=
5
𝑥
3
−4
𝑦=
4
𝑥
7
Direct Variation
• Direct Variation is a relationship that can be written in the form of:
y=kx, where k≠0
Where k is called the Constant of Variation
• k is also the coefficient of x
• When x and y are variables with coefficients, find k by solving for y
• When we know values to plug in for x and y, find k by plugging in the values
and solving for k
Direct Variation
• For a set of data to show direct variation, every set of x and y values
must produce the same constant of variation
• Even if the data produces a linear function, if the constant of variation
differs between coordinate pairs, it is not a direct variation
Direct Variation
Warm Up for 1/9
• Write the equation for each line in slope intercept form
Slope Intercept Form
• How do we find the slope?
• How do we find the y-intercept?
Slope Intercept Form
• What is the slope intercept equation for the line through points (0,4)
and (3, 10)?
Warm Up for 1/10
Find the equations in slope-intercept form for the lines that
pass through the following pairs of points:
• (0,7) & (2,4)
• (4,1) & (0,4)
• (2,-1) (4,7)
• (-5,-3) & (3,1)
Write the equations for the lines through the pairs of points
• (4,8) & (2,-1)
• (8,7) & (3,10)
Point Slope Form
• Sometimes an equation will have a y-intercept that is not a whole
number value
• In order to graph these accurately, we need a better way to write the
equation
• One way we can use is called Point Slope Form
Point-Slope Form
Point-Slope Form
Point-Slope Form
Point-Slope Form
• Write the equation for the line
graphed to the right
Point Slope Form
• Graph the Equation:
Point Slope Form
• Graph the Equation:
Warm Up for 1/11
Write the equations for the lines through the pairs of points
• (4,8) & (2,-1)
• (8,7) & (3,10)
Point-Slope Form
Write the equations for the lines through the pairs of points in both
point-slope and slope-intercept forms
• (1,5) & (7,-1)
• (4,7) & (6,10)
Point-Slope Form
Point-Slope Form
Warm Up for 1/12
• At a movie theater, a large bag of popcorn costs $8 and a large soda costs $4. A
group of friends spent $64 together. What are some possible combinations of
number of bags of popcorn and sodas they can purchase? Graph some of these
possibilities.
Intercepts
• When using slope-intercept form or y=mx+b, we use b to symbolize
the y-intercept
• What is the y-intercept?
Intercepts
• Based on the definition of the y-intercept, what is the x-intercept?
Intercepts
• Y-Intercept
• When the point is on the
• When x=
• Example Point:
• X-Intercept
• When the point is on the
• When y=
• Example Point:
Intercepts
• Using the intercepts from the
last slide, we can connect the
two points to form a graph of a
line
• Because we can do this, another
way we can find and equation
for a line is by using both
intercepts
Standard Form of a Line
• Standard form is:
• Ax + By = C
• Where A, B, and C are all real numbers and A and B are not both zero
• We can use standard form to quickly find the x and y intercepts
• Once we have the two points, we can complete the graph of the line
Standard Form
• What are the x- and y- intercepts of the graph of 3x + 4y = 24?
Standard Form
Standard Form
• Graph 3x + 4y = 24
Standard Form
• Graph x - 2y = -2
Standard Form
• Graph 2x + 5y = 20
Graph the following equations in Standard Form
• 6x + 8y = 24
• 3x – 9y = 27
Standard Form
Standard Form
Standard Form
Write each equation in standard form using integers.
•y=x4
• y  4 = 5(x  8)
Standard Form
Write each equation in standard form using integers.
Standard Form
Write each equation in standard form using integers.
Standard Form
Write each equation in standard form using integers.
Exit Card
1. Sf
2. Sfds
3. sd
Warm Up for 1/17
• For each set of points: a) write the point-slope equation of the line
through them, and b) write the equation in standard form
Modeling with Slope Intercept Form
• A car is traveling at 45 mi/h. Write an equation that models the total distance d
traveled after h hours.
• Suppose you are doing a 5000-piece puzzle. You have already placed 175 pieces. Every
minute you place 10 more pieces.
• Write an equation in slope-intercept form to model the number of pieces
placed.
• After 50 more minutes, how many pieces will you have placed?
Modeling with Slope Intercept Form
• Hudson is already 40 miles away from home on his drive back to college. He is
driving 65 mi/h. Write an equation that models the total distance d travelled after
h hours.
Modeling with Slope Intercept Form
• When Phil started his new job, he owed the company $65 for his uniforms. He is
earning $13 per hour. The cost of his uniforms is withheld from his earnings.
Write an equation that models the total money he has m after h hours of work.
Modeling with Direct Variation
• An equilateral triangle is a triangle with three equal sides. The perimeter of an
equilateral triangle varies directly with the length of one side. What is an
equation that relates the perimeter p and length l of a side?
Modeling with Direct Variation
• The amount a you fill a tub varies directly with the amount of time t you fill it.
Suppose you fill 25 gallons in 5 minutes. What is an equation that relates a and t?
Modeling with Standard Form
• At a movie theater, a large bag of popcorn costs $8 and a large soda costs $4. A
group of friends spent $64 together. Model this using an equation.
Modeling with Standard Form
Modeling with Standard Form
• Mike was the kicker for the football team. He scored 56 points during the season
kicking field goals (3 points) and extra points (1 point). Write an equation that
represents this situation and find three solutions.
Warm Up for 1/18
• You have only nickels and dimes in your piggy bank. When you ran the
coins through a change counter, it indicated you have 595 cents.
Write and graph an equation that represents this situation. What are
three combinations of nickels and dimes you could have?
• What if we also had quarters in the bank, what would the new
equation be?
Modeling with Standard Form
• You have $25 to buy supplies for a class party. Juice costs $3 per bottle and chips
cost $2 per bag. Write an equation that relates the amount of juice and chips you
can buy using $25. List some possible solutions.
Modeling with Standard Form
• You work two jobs. At the first job, you earn $10 per hour. At the second job, you
earn $12 per hour. You earned $440 last week. Write an equation that represents
this situation.
Modeling with Standard Form
• Juan can ride his bike at 12 mi/h and walk at 4 mi/h. Write an
equation that relates the amount of time he can spend riding or
walking combined, to travel 20 miles.