A linear function, as it is called, is an algebraic expression that represents the graph of a straight line.
When we talk about functions, it's important to highlight that the graphs of functions are represented in an axis system where there is a horizontal axis X and a vertical axis Y.
Linear functions can be expressed by the expressions y=mx or y=mx+b, where m represents the slope of the line while b (when it exists) represents the y-intercept.
To plot a linear function, all we need are 2 points. If the linear function is given, you can substitute a value for X and obtain the corresponding Y value.
We are asked to graph it on the coordinate system.
As we have discussed, to do this we need two points, which we will place in the function's expression. Choose any two points we like, it doesn't matter.
Now we will plot the two points on the coordinate system and connect them. This is actually a graph of the function for y=2x+1.
Examples and Exercises with Solutions for Linear Functions
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Calculate the slope of the line that passes through the points (4,1),(2,5).
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Remember the formula for calculating a slope using points:
Now, replace the data in the formula with our own:
(2ā4)(5ā1)ā=ā24ā=ā2
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-2
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What is the slope of a straight line that passes through the points (0,0),(ā8,2)?
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To solve the problem, remember the formula to find the slope using two points
Ā
Now, we replace the given points in the calculation:
Ā (0ā(ā8)(0ā2)ā=8ā2ā=ā41ā
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ā41ā
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Choose the correct answer for the function.
y=āx+1
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Let's start with option A
In a linear function, to check if the functions are parallel, you must verify if their slope is the same.
y = ax+b
The slope is a
In the original formula:
Ā y = -x+1
The slope is 1
In option A there is no a at all, which means it equals 1, which means the slope is not the same and the option is incorrect.
Ā
Option B:
To check if the function passes through the points, we will try to place them in the function:
-1 = -(-2)+1
-1 = 2+1
-1 = 3
The points do not match, and therefore the function does not pass through this point.
Ā
Option C:
We rearrange the function, in a way that is more convenient:
y = -1-x
y = -x-1
You can see that the slope in the function is the same as we found for the original function (-1), so this is the solution!
Ā
Option D:
When the slope is negative, the function is decreasing, as the slope is -1, the function is negative and this answer is incorrect.
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The graph is parallel to the graph of function
y=ā1āx
Test your knowledge
Question 1
Given the two tables of values x and and.
These tables represent a linear function. Fit an equation of a linear function to each one.