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Linear function and slopes of a line
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Transcript of Linear function and slopes of a line
Linear Functions
linear equations, intercepts and slopes
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
47
3
x y
x + 6y = 3
The equation is in the standard form
4x - y = 21
Examples of Nonlinear Equations
4x2 + y = 5
xy + x = 5
s/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.
32 xy
xy 25
652 yxy
354
1 yx
2x + y = 5
This is not a linear equation since its in the second degree
This is not a linear equation since variables are multiplied
x + 20y = 12
DEFINITION OF A LINEAR FUNCTION
A linear function is a function of the form
f(x) = mx + b
where m and b are real numbers and m = 0
Transform the following into the form y = mx + b
x + y = 2
2x – y = 5
8x – 2y = 12
-3x + 2y = 6
y = -x + 2
y = 2x - 5
y = 4x - 6
y = 3x + 3 2
Slope refers to the steepness of a line.
SLOPE OF A LINE
Slopes: trends
An increasing line defines a positive slope
A decreasing line defines a negative slope
A horizontal line defines a zero slope
A vertical line defines an undefined slope
Finding the slope of Linear functions
What is the slope of a line passes through points (4,6) and (3,4)?
m = 2
Determine the Slope of the following linear functions that passes through the given pair of points
1. (3, 2), (6, 6)
2. (-9, 6), (-10, 3)
3. (-4, 2), (-5, 4)
x and y intercepts
The 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 = 0
3x + 2y = 93x + 2(0) = 9 Replace y with 0
3x = 9 Divide each side by 3x = 3
To find the y- intercept, let x = 0
3x + 2y = 9
Replace x with 0
Divide each side by 2
3(0) + 2y = 9
2y = 9 y = 9/2
Plot the two points and connect them to draw the line.
2x + y = 4
To find the x- intercept, let y = 0Original Equation
Replace y with 0
2x + y = 4
2x + (0) = 4
2x =4 Divide each side by 3
x = 2
To find the y- intercept, let x = 02x + y = 4
2(0) + y = 4
y = 4
Original Equation
Replace x with 0
Simplify
Plot the two points and connect them to draw the line.
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 - 5
0 = 4y - 5
5 = 4y
= y
● (0, ) is the
y-intercept
5
4
5
4
Find the x and y-interceptsof g(x) = -3x – 1*
● x-intercept● ( , 0) is the
x-intercept
● y-intercept● (0, -1) is the
y-intercept
*g(x) is the same as y
1
3
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
●There is no y.
● 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
EQUATION OF A LINEAR FUNCTION
Slope- Intercept form
y = mx + b
y = mx + b
Give the equation of the linear function y in slope intercept form given its slope and y-intercept
1. m = -3, b = 2
2. m= 2, b = - 4
3. M = 1/3, b = 3
EQUATION OF A LINEAR FUNCTION
Point-Slope form
y –y1= m(x – x1)
y –y1= m(x – x1)
Give the equation of the linear function y with the given slope and passing through given points.
1. m = 2, through (1, 2)
2. m= -3, through (5, 0)
3. m = -1/3, through (-1, 3)
EQUATION OF A LINEAR FUNCTION
Give the equation of the linear function y with the given slope and passing through given points.
1. through (1, 2) and (3, -2)
2. through (5, 0) and (-1, 3)
Intercept Form
_x_ + _y_
a b
EQUATION OF A LINEAR FUNCTION
= 1
Questions??