3-2 Linear Functions
What is the difference between a linear function and a linear equation?
A function always has 2 or more variables, while an equation may have 0, 1, or more variables. A linear function graph is a straight line that can not be vertical; Not all linear equations form a straight line. In a linear equation, each term is either a constant or the product of a constant and a single variable.
linear function
A function in which the graph of the solutions forms a line; The function has constants and simple variables.
linear equation
An equation whose graph is a line
Consider the function f(x) = 3 (x-1) - 0.4 (9-x). Match each equation with its equivalent value: I. f(2) + f(4) II. f(5) III. f(7) - f(6) IV. f(3) A. 3.4 B. 3.6 C. 7.2 D. 10.4
I. matches with C II. matches with D III. matches with A IV. matches with B
A chairlift starts 0.5mi above the base of a mountain at a constant speed of 6 miles per hour. It takes 15 minutes to reach the top. Write a function for this situation. How far from the base of the mountain is the chairlift after 10 minutes? How would the function, graph and equation change if the speed is 4mph?
Must convert miles per hour to miles per min = 0.1 miles/min Distance traveled = rate of chairlift x time traveling + initial distance from base d(t) = 0.1 t + 0.5 After 10 minutes, the chair lift would be 1.5 miles up the mountain The speed would be 0.06 repeating which would be the new slope of the graph making it less steep.
Is the function linear? xy + 7 = x + y
No - not linear
Is the function linear? x² + xy = 7
No - not linear
The cost of using a game facility is $1 for every 12 minutes. Talisa writes the function for the cost per hour as f(x) = 12x. Is she right? Why?
No, if $1 per 12 minutes; then concert to per hour would be $5. Therefore function would be f(x) = 5x
Is the function linear? x + 3y² = 6
No- not linear
A periscope is 24 inches above the surface. It ascends at 6 inches per second. What function models the height of the periscope lens at time t? If the periscope reaches its maximum height after ascending for 22 seconds, what is the maximum height in feet?
Rate is 6 inches per second f(x) = 6t height at 22 would be: f(22) = 132 inches
The cost to make 4 bracelets is shown as : # bracelets: 1 , 2 , 3 , 4 cost: 17, 32, 47, 62 What is the relationship between the values?
The constant rate of change is 1 and 15
Is the function linear? 2x - 5y + 8 = 0
Yes - linear
Is the function linear? x + 3y = 7
Yes - linear
Is the function linear? x= -2
Yes - linear
Is the function linear? y = 2x + 3
Yes - linear
function notation
an equation in the form of 'f(x)=' to show the output value of a function, f, for an input value x
Evaluate each function for x = 2 and x = 6 f(x) = - (x - 2)
f(2) = 0 f(6) = -4
Evaluate each function for x = 2 and x = 6 f(x) = 4x - 3
f(2) = 5 f(6) = 21
Find f(5) for f(x) = -2 (x+1)
f(5) = -21
Find f(5) for f(x) = 6 + 3x
f(5) = 21
Find f(5) for f(m) = 2(m-3)
f(5) = 4
The cost to make 4 bracelets is shown as : # bracelets: 1 , 2 , 3 , 4 cost: 17, 32, 47, 62 Find the value of b in the slope intercept form?
f(x) = 15 (x) + b 17 = 15 (1) + b 2 = b
The cost to make 4 bracelets is shown as : # bracelets: 1 , 2 , 3 , 4 cost: 17, 32, 47, 62 Write a function to determine the cost of any number of bracelets?
f(x) = 15 x + 2
The cost to make 4 bracelets is shown as : # bracelets: 1 , 2 , 3 , 4 cost: 17, 32, 47, 62 Write a function using slope intercept form?
f(x) = 15 x + b
Tomeka records the outside temperature at 6:00am, to be -3 degrees F. The outside temperature increases by 2 degrees F every hour for the next 6 hours. If the temperature continues to rise at the same rate: Write a function for this situation. What will the temperature be at 2:00 pm? Is this a linear function? Is this linear function realistic for the domain 0<x<24?
f(x) = 2x - 3 Therefore the temperature at 2pm would be 13 degrees F. Yes No, the temperature will likely drop in the evening.
Evaluate the function for x=4: g(x) = -2x - 3
g(4) = -11
Evaluate the function for x=4: h(x) = 7x + 15
h(4) = 43
Consider the two functions: g(x) = 2x + 1 h(x) = 2x + 2 How do the ranges compare?
h(x) is g(x) +1 therefore the range will always be 1 unit greater than that of f(x).
Write a linear function for the data in the table using function notation: X: 1 , 2 , 3 , 4 Y: 6.5, 13 , 19.5 , 26
m= (y2 - y1) / (x2 - x1) m = (13-6.5) / (2-1) m = 6.5 / 1 = 6.5 y-y = m (x - x1) y - 6.5 = 6.5 (x - 1) y= 6.5x - 6.5 + 6.5 y = 6.5 x f(x) = 6.5x
Write a linear function for the data in the table using function notation: X: 1 , 2 , 3 , 4 Y: 1 . 4 . 7 . 10
m= (y2 - y1) / (x2 - x1) m = (4-1) / (2-1) m = 3 / 1 = 3 y-y = m (x - x1) y - 1 = 3 (x - 1) y= 3x - 3 + 1 y = 3x - 2 f(x) = 3x - 2
Write a linear equation for the data: X: -2, -1, 0, 1, 2 Y: 2, 0.5, -1, -2.5, -4
m= (y₂₋y₁) / (X₂₋X₁) m = -1.5 y = mx + b y = -1.5x + -1 = f(x)
The following 2 points are on a graph: (1,2) and (4,1) Write a function to represent the graph. What is f(6)?
m= (y₂₋y₁) / (X₂₋X₁) m = -1/3 y-y₁ = m (x-x₁) y-2 = - (1/3) (x-1) y = -(1/3)x + (7/3) = f(x) f(6) = 1/3
Write a linear function for the data: X: 0, 1, 2, 3, 4 Y: 4, 1.5, -1, -3.5, -6
m= (y₂₋y₁) / (X₂₋X₁) m = -2.5 y = mx + b y = -2.5x + 4 = f(x)
Write a linear function for the data: X: 0, 1, 2, 3, 4 Y: -1, 4, 9, 14, 19
m= (y₂₋y₁) / (X₂₋X₁) m = 5 y = mx +b y = 5x + (-1) y = 5x - 1 = f(x)