Linear Functions
TLW identify linear equations and intercepts.
A linear equation is the equation of a line.The standard form of a linear equation is
Ax + By = C* A has to be positive and cannot be a
fraction.
Examples of linear equations
2x + 4y =8
6y = 3 – x
x = 1
-2a + b = 5
4
73
x y
Equation is in Ax + By =C form
Rewrite with both variables on left side … x + 6y =3
B =0 … x + 0 y =1
Multiply both sides of the equation by -1 … 2a – b = -5
Multiply both sides of the equation by 3 … 4x –y =-21
Examples of Nonlinear Equations
4x2 + y = 5 xy + x = 5s/r + r = 3
The exponent is 2
There is a radical in the equation
Variables are multiplied
Variables are divided
4x
The following equations are NOT in the standard form of Ax + By =C:
Determine whether the equation is a linear equation, if so write it in standard form.y = 5 – 2x
y = 5 – 2x Rewrite the equation
+ 2x+ 2x Add 2x to each side
2x + y = 5 Simplify
A = 2, B= 1, C=5 This IS a linear equation.
2xy -5y = 6
Determine whether the equation is a linear equation, if so write it in standard form.
Since the term 2xy has two variables, the equation cannot be written in the form Ax + By =0. Therefore, this is NOT a linear equation.
Determine whether the equation is a linear equation, if so write it in standard form.
32 xy
Since the term x is raised to the second power, the equation cannot be written in the form Ax + By =0. Therefore, this is NOT a linear equation.
y = 6 – 3x
Determine whether the equation is a linear equation, if so write it in standard form.
Rewrite the equation
y = 6 – 3x Add 3x to each side+ 3x+ 3x
Simplify3x +y = 6
A = 3, B= 1, C=6This IS a linear equation.
Determine whether the equation is a linear equation, if so write it in standard form.
3541
yx
Multiply everything by the denominator to get rid of the fraction
)3541( yx(4)
x + 20y = 12
A = 1, B= 20, C=12This IS a linear equation.
-4x+7=2
Determine whether the equation is a linear equation, if so write it in standard form.
X and Y interceptsThe x coordinate of the point at which the graph of an
equation crosses the x –axis is the x- intercept .The y coordinate of the point at which the graph of an
equation crosses the y-axis is called the y- intercept.
X- intercept (-x,0)
y- intercept (0, y)
3x + 2y = 9
Graph the linear equation using the x- intercept and the y intercept
To find the x- intercept, let y = 03x + 2y = 9 Original Equation
3x + 2(0) = 9 Replace y with 03x = 9 Divide each side by 3
x = 3
To find the y- intercept, let x = 03x + 2y = 9 Original Equation
Replace x with 0
Divide each side by 2
3(0) + 2y = 9
2y = 9 y = 4.5
Plot the two points and connect them to draw the line.
2x + y = 4
Graph the linear equation using the x- intercept and the y intercept
To find the x- intercept, let y = 0Original Equation
Replace y with 0
2x + y = 42x + (0) = 4
2x =4 Divide each side by 3
x = 2To find the y- intercept, let x = 0
2x + y = 42(0) + y = 4
y = 4
Original Equation
Replace x with 0
Simplify
Plot the two points and connect them to draw the line.
Identify the x- and y- intercepts given a table
X Y
-1 -6
0 -4
1 -2
2 0
3 2
X Y
-4 1
-3 0
-2 -1
-1 -2
0 -3
Find the x and y- interceptsof x = 4y – 5
● x-intercept:● Plug in y = 0
x = 4y - 5 x = 4(0) - 5 x = 0 - 5 x = -5● (-5, 0) is the x-intercept
● y-intercept:● Plug in x = 0
x = 4y - 50 = 4y - 55 = 4y
= y
● (0, ) is the y-intercept
54
54
Find the x and y-interceptsof g(x) = -3x – 1*
● x-intercept● Plug in y = 0
g(x) = -3x - 1 0 = -3x - 1 1 = -3x
= x● ( , 0) is the x-intercept
● y-intercept● Plug in x = 0
g(x) = -3(0) - 1g(x) = 0 - 1g(x) = -1
● (0, -1) is the y-intercept
*g(x) is the same as y
13
13
Find the x and y-intercepts of 6x - 3y =-18
● x-intercept● Plug in y = 0
6x - 3y = -18 6x -3(0) = -18 6x - 0 = -18 6x = -18 x = -3
● (-3, 0) is the x-intercept
● y-intercept● Plug in x = 0
6x -3y = -18 6(0) -3y = -18 0 - 3y = -18 -3y = -18 y = 6
● (0, 6) is the y-intercept
Find the x and y-intercepts of x = 3
● y-intercept
● A vertical line never crosses the y-axis.
● There is no y-intercept.
● x-intercept
● Plug in y = 0.
There is no y. Why?
● x = 3 is a vertical line so x always equals 3.
● (3, 0) is the x-intercept.
x
y
Find the x and y-intercepts of y = -2
● x-intercept
● Plug in y = 0. y cannot = 0 because y = -2.● y = -2 is a horizontal line so it never crosses the x-axis.
●There is no x-intercept.
● y-intercept
● y = -2 is a horizontal line
so y always equals -2.
● (0,-2) is the y-intercept.
x
y
Graph by making a tableGraph 3
21
xy
Select values from the domain and make a table. Then graph the order pairs. Draw a line through the points
x y (x, y)-2 3)2(
21
-4 (-2, -4)0 3)0(
21
-3 (0, -3)
2 3)2(21
-2 (2, -2)
321
x
Graph by making a tableGraph
x y (x, y)
Select values from the domain and make a table. Then graph the order pairs. Draw a line through the points
2 xy
2 xy
Questions??