Variation of a Function Practice Problems and Exercises

Master function variation, rate of change, and slope concepts with step-by-step practice problems. Learn to identify coefficients and analyze function behavior.

📚What You'll Master in Function Variation Practice
  • Identify coefficients a, b, and c in quadratic and linear functions
  • Determine when functions are positive or negative using graphs
  • Calculate the rate of change and slope of various functions
  • Analyze constant vs. variable rates of change in different function types
  • Interpret intersection points and their significance in function behavior
  • Apply function variation concepts to solve real-world problems

Understanding Variation of a Function

Complete explanation with examples

The variation of a function means the rate at which a certain function changes. The rate of variation of a function is also called the slope.

According to the mathematical definition, the slope represents the change of the function (Y) (Y) by increasing the value of X X by 1 1 .

  • If the function's graph is represented by a straight line, it means that the rate of variation of the function is constant
  • However, if the graph is not represented by a straight line, this implies that the rate of variation of the function is not constant
Detailed explanation

Practice Variation of a Function

Test your knowledge with 9 quizzes

Given a table showing points on the graph of a function, determine whether or not the rate of change is uniform.

XY1234-6-303

Examples with solutions for Variation of a Function

Step-by-step solutions included
Exercise #1

Look at the graph below and determine whether the function's rate of change is constant or not:

–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777888999101010111111–3–3–3–2–2–2–1–1–1111222333444555000

Step-by-Step Solution

First we need to remember that if the function is not a straight line, its rate of change is not constant.

The rate of change is not uniform since the function is not a straight line.

Answer:

Not constant

Video Solution
Exercise #2

Given the following graph, determine whether the rate of change is uniform or not

111222333444555666777888999101010111111121212131313141414151515111222333444555666777888000

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:

Non-uniform

Video Solution
Exercise #3

Given the following graph, determine whether the rate of change is uniform or not?

–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777888999101010111111–3–3–3–2–2–2–1–1–1111222333444555000

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:

Non-uniform

Video Solution
Exercise #4

Given the following graph, determine whether the rate of change is uniform or not?

–8–8–8–7–7–7–6–6–6–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777888–3–3–3–2–2–2–1–1–1111222333444555666000

Step-by-Step Solution

Remember that if the function is a straight line, its rate of change will be constant.

Due to the fact that the graph is a straight line - the rate of change is constant.

Answer:

Uniform

Video Solution
Exercise #5

Given the following graph, determine whether the rate of change is uniform or not

–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777888999101010111111–3–3–3–2–2–2–1–1–1111222333444555000

Step-by-Step Solution

The problem asks us to determine if the rate of change in the graph is uniform or not. To do this, we need to examine the graph closely to see whether it is linear.

If a graph is linear, it means it is a straight line, indicating a constant (uniform) rate of change. The slope of a straight line does not change, meaning that for every unit increase in x x there is a proportional and consistent change in y y .

In contrast, if a graph curves or the line is not straight, the rate of change would not be uniform. This is because a curve indicates that the amount y y changes for each unit change in x x is not constant.

By analyzing the given graph, we can see that it is a non-linear function with a visible curve. Since the line is not straight (it appears as a curved line in the graph), the rate of change of the function is not constant across its range.

Therefore, the solution to the problem is that the rate of change is non-uniform.

Consequently, the correct choice, corresponding to a non-uniform rate of change in the graph, is:

Non-uniform

Answer:

Non-uniform

Video Solution

Frequently Asked Questions

Everything you need to know about Variation of a Function

What is the variation of a function in simple terms?

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The variation of a function is the rate at which the function changes, also known as the slope. It represents how much the function's y-value changes when the x-value increases by 1 unit.

How do I identify coefficients a, b, and c in a quadratic function?

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In a quadratic function y = ax² + bx + c: 'a' is the coefficient of x², 'b' is the coefficient of x, and 'c' is the constant term (free number). If any term is missing, its coefficient is 0.

When is a function positive or negative on a graph?

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A function is positive when its graph is above the x-axis (y > 0) and negative when below the x-axis (y < 0). The function equals zero at intersection points with the x-axis.

What's the difference between constant and variable rate of change?

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Constant rate of change occurs in linear functions (straight line graphs) where the slope remains the same throughout. Variable rate of change occurs in non-linear functions where the slope changes at different points.

How do I find where a linear function intersects the x-axis?

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To find x-intercepts, set y = 0 and solve for x. For example, in y = -40x + 40, setting y = 0 gives 0 = -40x + 40, so x = 1. The intersection point is (1, 0).

Why is understanding function variation important in math?

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Function variation helps you: 1) Predict function behavior, 2) Solve optimization problems, 3) Understand rates in real-world scenarios, 4) Prepare for calculus concepts like derivatives.

What are common mistakes when identifying function coefficients?

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Common errors include: forgetting that missing terms have coefficient 0, confusing the order of coefficients, and not recognizing negative signs as part of the coefficient.

How do I determine if a quadratic function opens upward or downward?

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Look at the coefficient 'a' in y = ax² + bx + c. If a > 0, the parabola opens upward. If a < 0, it opens downward. This affects where the function is positive or negative.

More Variation of a Function Questions

Click on any question to see the complete solution with step-by-step explanations

Variation of a Function

Find the Rate of Change: Solving -x+y=0 Linear Equation Find the Rate of Change in the Linear Equation: -5x+y=3 Find the Rate of Change in the Linear Equation 3-y=1/4x Find the Rate of Change in Linear Equation -8y+2x=6 Find the Rate of Change: Solving -1/4y + x = 3

Continue Your Math Journey

Master Variation of a Function first, then advance to these related topics that build on your fraction skills

Suggested Topics to Practice in Advance

Ways to Represent a FunctionRepresenting a Function Verbally and with TablesGraphical Representation of a FunctionAlgebraic Representation of a FunctionNotation of a Function

Topics Learned in Later Sections

Rate of Change of a FunctionRate of change represented with steps in the graph of the functionRate of change of a function represented graphicallyConstant Rate of ChangeVariable Rate of ChangeRate of Change of a Function Represented by a Table of ValuesFunctions for Seventh GradeIncreasing and Decreasing Intervals (Functions)Increasing functionsDecreasing functionConstant FunctionDecreasing Interval of a functionIncreasing Intervals of a functionDomain of a FunctionIndefinite integralInputing Values into a Function

Practice by Question Type

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Calculate the rate of change from an equation Calculate the rate of change from a table Calculate the rate of change in a graph Drawing conclusions from a table or graph