4.1 Linear functions  (Page 8/27)

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The other characteristic of the linear function is its slope .

Let’s consider the following function.

$f\left(x\right)=\frac{1}{2}x+1$

The slope is $\text{\hspace{0.17em}}\frac{1}{2}.\text{\hspace{0.17em}}$ Because the slope is positive, we know the graph will slant upward from left to right. The y- intercept is the point on the graph when $\text{\hspace{0.17em}}x=0.\text{\hspace{0.17em}}$ The graph crosses the y -axis at $\text{\hspace{0.17em}}\left(0,1\right).\text{\hspace{0.17em}}$ Now we know the slope and the y -intercept. We can begin graphing by plotting the point $\text{\hspace{0.17em}}\left(0,1\right).\text{\hspace{0.17em}}$ We know that the slope is the change in the y -coordinate over the change in the x -coordinate. This is commonly referred to as rise over run, $\text{\hspace{0.17em}}m=\frac{\text{rise}}{\text{run}}.\text{\hspace{0.17em}}$ From our example, we have $\text{\hspace{0.17em}}m=\frac{1}{2},$ which means that the rise is 1 and the run is 2. So starting from our y -intercept $\text{\hspace{0.17em}}\left(0,1\right),$ we can rise 1 and then run 2, or run 2 and then rise 1. We repeat until we have a few points, and then we draw a line through the points as shown in [link] .

Graphical interpretation of a linear function

In the equation $\text{\hspace{0.17em}}f\left(x\right)=mx+b$

• $b\text{\hspace{0.17em}}$ is the y -intercept of the graph and indicates the point $\text{\hspace{0.17em}}\left(0,b\right)\text{\hspace{0.17em}}$ at which the graph crosses the y -axis.
• $m\text{\hspace{0.17em}}$ is the slope of the line and indicates the vertical displacement (rise) and horizontal displacement (run) between each successive pair of points. Recall the formula for the slope:

Do all linear functions have y -intercepts?

Yes. All linear functions cross the y-axis and therefore have y-intercepts. (Note: A vertical line is parallel to the y-axis does not have a y-intercept, but it is not a function .)

Given the equation for a linear function, graph the function using the y -intercept and slope.

1. Evaluate the function at an input value of zero to find the y- intercept.
2. Identify the slope as the rate of change of the input value.
3. Plot the point represented by the y- intercept.
4. Use $\text{\hspace{0.17em}}\frac{\text{rise}}{\text{run}}\text{\hspace{0.17em}}$ to determine at least two more points on the line.
5. Sketch the line that passes through the points.

Graphing by using the y- Intercept and slope

Graph $\text{\hspace{0.17em}}f\left(x\right)=-\frac{2}{3}x+5\text{\hspace{0.17em}}$ using the y- intercept and slope.

Evaluate the function at $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ to find the y- intercept. The output value when $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ is 5, so the graph will cross the y -axis at $\text{\hspace{0.17em}}\left(0,5\right).$

According to the equation for the function, the slope of the line is $\text{\hspace{0.17em}}-\frac{2}{3}.\text{\hspace{0.17em}}$ This tells us that for each vertical decrease in the “rise” of $\text{\hspace{0.17em}}–2\text{\hspace{0.17em}}$ units, the “run” increases by 3 units in the horizontal direction. We can now graph the function by first plotting the y -intercept on the graph in [link] . From the initial value $\text{\hspace{0.17em}}\left(0,5\right)\text{\hspace{0.17em}}$ we move down 2 units and to the right 3 units. We can extend the line to the left and right by repeating, and then drawing a line through the points.

Find a point on the graph we drew in [link] that has a negative x -value.

Possible answers include $\text{\hspace{0.17em}}\left(-3,7\right),\text{\hspace{0.17em}}$ $\left(-6,9\right),\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\left(-9,11\right).$

Graphing a function using transformations

Another option for graphing is to use a transformation of the identity function $\text{\hspace{0.17em}}f\left(x\right)=x.\text{\hspace{0.17em}}$ A function may be transformed by a shift up, down, left, or right. A function may also be transformed using a reflection, stretch, or compression.

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
More example of algebra and trigo
What is Indices
If one side only of a triangle is given is it possible to solve for the unkown two sides?
cool
Rubben
kya
Khushnama
please I need help in maths
Okey tell me, what's your problem is?
Navin