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4.1 Linear functions  (Page 10/27)

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y = 2 x + 4

Given a graph of linear function, find the equation to describe the function.

  1. Identify the y- intercept of an equation.
  2. Choose two points to determine the slope.
  3. Substitute the y- intercept and slope into the slope-intercept form of a line.

Matching linear functions to their graphs

Match each equation of the linear functions with one of the lines in [link] .

a .   f ( x ) = 2 x + 3 b . g ( x ) = 2 x − 3 c . h ( x ) = −2 x + 3 d .  j ( x ) = 1 2 x + 3
Graph of four functions where the orange line has a y-intercept at 3 and slope of 2, the baby blue line has a y-intercept at 3 and slope of 1/2, the blue line has a y-intercept at 3 and slope of -2, and the green line has a y-intercept at -3 and slope of 2.

Analyze the information for each function.

  1. This function has a slope of 2 and a y -intercept of 3. It must pass through the point (0, 3) and slant upward from left to right. We can use two points to find the slope, or we can compare it with the other functions listed. Function   g   has the same slope, but a different y- intercept. Lines I and III have the same slant because they have the same slope. Line III does not pass through   ( 0 , 3 )   so   f   must be represented by line I.
  2. This function also has a slope of 2, but a y -intercept of   −3.   It must pass through the point   ( 0 , −3 )   and slant upward from left to right. It must be represented by line III.
  3. This function has a slope of –2 and a y- intercept of 3. This is the only function listed with a negative slope, so it must be represented by line IV because it slants downward from left to right.
  4. This function has a slope of   1 2   and a y- intercept of 3. It must pass through the point (0, 3) and slant upward from left to right. Lines I and II pass through   ( 0 , 3 ) , but the slope of   j   is less than the slope of   f   so the line for   j   must be flatter. This function is represented by Line II.

Now we can re-label the lines as in [link] .

Graph of four functions where the blue line is h(x) = -2x + 3 which goes through the point (0,3), the baby blue line is j(x) = x/2 + 3 which goes through the point (0,3). The orange line is f(x) = 2x + 3 which goes through the point (0,3), and the red line is g(x) = 2x – 3 which goes through the point (0,-3).
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Finding the x -intercept of a line

So far we have been finding the y- intercepts of a function: the point at which the graph of the function crosses the y -axis. Recall that a function may also have an x -intercept , which is the x -coordinate of the point where the graph of the function crosses the x -axis. In other words, it is the input value when the output value is zero.

To find the x -intercept, set a function   f ( x )   equal to zero and solve for the value of   x .   For example, consider the function shown.

f ( x ) = 3 x − 6

Set the function equal to 0 and solve for   x .

0 = 3 x − 6 6 = 3 x 2 = x x = 2

The graph of the function crosses the x -axis at the point   ( 2 , 0 ) .

Do all linear functions have x -intercepts?

No. However, linear functions of the form   y = c , where   c   is a nonzero real number are the only examples of linear functions with no x-intercept. For example,   y = 5   is a horizontal line 5 units above the x-axis. This function has no x-intercepts, as shown in [link] .

Graph of the function y = 5, a completely horizontal line that goes through the point (0,5). Graphed on an xy-plane with the x-axis ranging from -3 to 3 and the y-plane ranging from -1 to 8.

x -intercept

The x -intercept of the function is value of   x   when   f ( x ) = 0.   It can be solved by the equation   0 = m x + b .

Finding an x -intercept

Find the x -intercept of   f ( x ) = 1 2 x − 3.

Set the function equal to zero to solve for   x .

0 = 1 2 x − 3 3 = 1 2 x 6 = x x = 6

The graph crosses the x -axis at the point   ( 6 , 0 ) .

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Find the x -intercept of   f ( x ) = 1 4 x − 4.

  ( 16 ,  0 )

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Describing horizontal and vertical lines

There are two special cases of lines on a graph—horizontal and vertical lines. A horizontal line    indicates a constant output, or y -value. In [link] , we see that the output has a value of 2 for every input value. The change in outputs between any two points, therefore, is 0. In the slope formula, the numerator is 0, so the slope is 0. If we use   m = 0   in the equation   f ( x ) = m x + b , the equation simplifies to   f ( x ) = b .   In other words, the value of the function is a constant. This graph represents the function   f ( x ) = 2.

Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
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Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
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Haryormhidey Reply
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ALIYU
field is a region of space under the influence of some physical properties
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WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
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Nassze Reply
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Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
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Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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