That is, there are functions, such as the linear function (which we will study in more detail later, but generally speaking, it is a function with the variable to the first power ) in which the slope, or in other words, the rate of change of the function is constant, and there are other functions that may have an increasing or decreasing rate of change that is calculated separately for each value X X X
Exercises on the variation of a function Exercise 1 Assignment
y = − 5 x 2 + x y=-5x^{2}+x y = − 5 x 2 + x
Solution
a a a x 2 x^2 x 2
Given in the exercise: − 5 -5 − 5
b b b x x x
Given in the exercise: 1 1 1
c c c
Therefore it is: 0 0 0
Answer
a = − 5 , b = 1 , c = 0 a=-5,~b=1,~c=0 a = − 5 , b = 1 , c = 0
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Exercise 2 Assignment
Given the linear function in the graph
When is the function positive?
Solution
The function is positive when it is above the axis: x x x
Pay attention that the intersection point with the axis x x x ( 2 , 0 ) \left(2,0\right) ( 2 , 0 )
According to the graph, the function is positive, therefore x > 2 x\gt2 x > 2
Answer
x > 2 x\gt2 x > 2
Exercise 3 Assignment
Given the function in the graph
When is the function positive?
Solution
The intersection point with the axis :x x x ( − 4 , 0 ) \left(-4,0\right) ( − 4 , 0 )
First positive, then negative.
Therefore x < − 4 x<-4 x < − 4
Answer
x < − 4 x<-4 x < − 4
Do you know what the answer is?
Exercise 4 Assignment
y = − 40 x + 40 y=-40x+40 y = − 40 x + 40
Solution
a a a x 2 x^2 x 2
Given in the exercise: 0 0 0
b b b x x x
Given in the exercise: − 40 -40 − 40
c c c
Therefore it is: 40 40 40
Answer
a = 0 , b = − 40 , c = 40 a=0,~b=-40,~c=40 a = 0 , b = − 40 , c = 40
Exercise 5 Assignment
y = − x 2 + 3 x + 40 y=-x^{2}+3x+40 y = − x 2 + 3 x + 40
Solution
a a a x 2 x^2 x 2
Given in the exercise: − 1 -1 − 1
b b b x x x
Given in the exercise: 3 3 3
c c c
Therefore it is: 40 40 40
Answer
a = − 1 , b = 3 , c = 40 a=-1,~b=3,~c=40 a = − 1 , b = 3 , c = 40
Examples with solutions for Variation of a Function Exercise #1 Given the following graph, determine whether the rate of change is uniform or not
–5 –5 –5 –4 –4 –4 –3 –3 –3 –2 –2 –2 –1 –1 –1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 11 11 11 –3 –3 –3 –2 –2 –2 –1 –1 –1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 0 0 0
Video Solution Step-by-Step Solution Let's remember that if the function is not a straight line, its rate of change is not uniform.
Since the graph is not a straight line - the rate of change is not uniform.
Answer Exercise #2 Given the following graph, determine whether the rate of change is uniform or not
–5 –5 –5 –4 –4 –4 –3 –3 –3 –2 –2 –2 –1 –1 –1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 11 11 11 –3 –3 –3 –2 –2 –2 –1 –1 –1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 0 0 0
Video Solution Step-by-Step Solution Let's remember that if the function is not a straight line, its rate of change is not uniform.
Since the graph is not a straight line - the rate of change is not uniform.
Answer Exercise #3 Given the following graph, determine whether the rate of change is uniform or not
1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 11 11 11 12 12 12 13 13 13 14 14 14 15 15 15 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 0 0 0
Video Solution Step-by-Step Solution Let's remember that if the function is not a straight line, its rate of change is not uniform.
Since the graph is not a straight line - the rate of change is not uniform.
Answer Exercise #4 Given the following graph, determine whether the rate of change is uniform or not
–8 –8 –8 –7 –7 –7 –6 –6 –6 –5 –5 –5 –4 –4 –4 –3 –3 –3 –2 –2 –2 –1 –1 –1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 –3 –3 –3 –2 –2 –2 –1 –1 –1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 0 0 0
Video Solution Step-by-Step Solution Remember that if the function is a straight line, its rate of change will be constant.
Since the graph is a straight line - the rate of change is constant.
Answer Exercise #5 Given the following graph, determine whether the rate of change is uniform or not
–5 –5 –5 –4 –4 –4 –3 –3 –3 –2 –2 –2 –1 –1 –1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 11 11 11 –3 –3 –3 –2 –2 –2 –1 –1 –1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 0 0 0
Video Solution Answer 
