Topics
2. Quadratic Functions & Equations
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United States of AmericaFL
Grade 912

2.08 Modeling with quadratics

Lesson
Practice
Lesson

Concept summary

Functions can be used to model real-world events and interpret data from those events. Data that measures or compares two characteristics of a population is known as bivariate data.

When analyzing and interpreting data, we often look for a relationship between two variables called an association or correlation.

Association

A way to describe the relationship between the two variables in a bivariate data set. For numerical data, descriptions include linear or nonlinear; positive or negative; strong or weak. For categorical data, descriptions include strong or weak.

Correlation

A relationship between two variables

xg\left(x\right)\text{first} \\ \text{difference}\text{second}\\ \text{difference}
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00-1-1-\left(-3\right)=2
1111-\left(-1\right)=2
2433-1=2
3955-3=2

We have previously seen that linear data have constant first differences.

The second difference is the difference between two consecutive first differences.

Data that have constant second differences are said to have a quadratic association.

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To more easily analyze a set of data and determine if there is a quadratic association between the variables, we often construct a scatter plot.

The different quadratic forms are useful for modeling different quadratic scenarios based on what information is given.

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The factored form of a quadratic function is f\left(x\right)=a\left(x-x_1\right)\left(x-x_2\right).

  • Useful when we have two points that can be represented as the x-intercepts
  • The x-intercepts often represent the points when a parabolic arc meets the ground
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The vertex form of a quadratic function is \\f\left(x\right)=a\left(x-h\right)^2+k

  • Useful when we know the maximum or minimum point of a quadratic situation

  • The x-coordinate of the vertex is where or when the maximum occurs

  • They-coordinate of the vertex is the minimum or maximum function value

Converting to standard form of a quadratic function can be useful if we need to know the y-intercept.

The curve of best fit and R^2 value can be calculated using technology to approximately model data. The value of R^2 can vary between 0 and 1. The closer the value is to 1 the more accurate the model is.

The model may only be appropriate over a part of the domain.

Domain constraint

A limitation or restriction of the possible x-values, usually written as an equation, inequality, or in set-builder notation

Worked examples

Example 1

The population, y, of Manatee that regularly visit a river is tracked over a number of years, x, (starting at zero) with the data displayed in the table:

x0123456
y605860667690108
a

Determine if the population represents a quadratic relation.

Approach

Construct a table to find the first and second difference.

If the second difference is constant for all x, and the first difference is not, then the relation is quadratic.

Solution

We get the following table of values:

xy\text{First difference}\text{Second difference}
060
158-2
26024
36664
476104
590144
6108184

The second difference is constant, so the function is quadratic.

b

Write an equation describing the relation shown in the table.

Approach

The table indicates that the vertex is at \left(1, 58\right) as this is the minimum value, so we will determine the equation in vertex form y=a\left(x-h\right)^2+k. We can then substitute the values of another point to solve for a.

Solution

The vertex is at \left(1, 58\right), which gives h=1, k=58. This means the equation is:y=a\left(x-1\right)^2+58 for some value of a. We can now substitute in the values of another point, such as \left(0, 60\right).

\displaystyle y\displaystyle =\displaystyle a\left(x-1\right)^2+58State the equation
\displaystyle 60\displaystyle =\displaystyle a\left(0-1\right)^2+58Substitute x=0, y=60
\displaystyle 60\displaystyle =\displaystyle a+58Simplify
\displaystyle 2\displaystyle =\displaystyle aSubtract 58

The equation that describes the relationship in the table is y=2\left(x-1\right)^2+58

Reflection

Writing this equation in standard form: y=2x^2-4x+60 shows that there is a y-intercept at \left(0, 60\right) which agrees with the value in the table.

c

Using the model in part (b), determine the population 10 years afer the numbers were first recorded.

Approach

We can find the population, y, after 10 years by substituing x=10 into the equation.

Solution

\displaystyle y\displaystyle =\displaystyle 2\left(x-1\right)^2+58State the equation
\displaystyle y\displaystyle =\displaystyle 2\left(10-1\right)^2+58Substitute x=10
\displaystyle y\displaystyle =\displaystyle 220Simplify

We can see that after 10 years the population will have grown to 220.

Example 2

Carlos is trying to determine the optimum angle he should kick a soccer ball from out of his hands to achieve the maximum distance. He records 10 kicks and analyses them to determine the angle of trajectory and also the distance travelled. His results are recorded in the table below:

Angle (degrees)24303337434851566064
Distance (feet)112129138155161164158148134124
a

Determine if the data suggests a quadratic association. Explain your answer.

Approach

We can determine if the data suggests a quadratic association by plotting the points on a coordinate plane and determining if the data resembles a parabola. We can also use technology to calculate the regression with a polynomial of degree 2. If the value of R^2 is close to 1 then there is likely to be a quadratic association.

Solution

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Yes, a quadratic model is appropriate because when drawn on a graph it has a parabolic shape which is symmetric.

Also, if we calculate regression on this data set with a polynomial of degree 2, we get R^2=0.9662 which is very close to 1, so there is a strong correlation to the quadratic model suggesting a quadratic association.

b

Using technology, determine an appropriate equation to model the data set to four decimal places.

Solution

y=-0.1132x^2+10.3245x-74.5885, where x is the angle in degrees and y is the distance travelled.

c

Calculate and interpret the meaning of the vertex of the model.

Approach

The vertex is the maximum height of the parabola and occurs on the line of symmetry, when x=-\dfrac{b}{2a}. We can then substitute this value into the equation found in part (b) to find y, the maximum distance.

Solution

The vertex represents the optimum angle to kick the ball to achieve the maximum distance travelled.

\displaystyle x\displaystyle =\displaystyle \frac{b}{2a}Line of symmetry
\displaystyle x\displaystyle =\displaystyle -\frac{10.3245}{2\left(-0.1132\right)}Substitute a=-0.1132, b=10.3245
\displaystyle x\displaystyle \approx\displaystyle 45.6Simplify

We can now substitute x=45.6 into the equation to solve for y.

\displaystyle y\displaystyle =\displaystyle -0.1132x^2+10.3245x-74.5885State the equation
\displaystyle y\displaystyle =\displaystyle -0.1132\left(45.6\right)^2+10.3245\left(45.6\right)-74.5885Substitute x=45.6
\displaystyle y\displaystyle \approx\displaystyle 160.8Simplify

The vertex is a maximum and occurs at about \left(45.6, 160.8\right) which means that the maximum distance of 160.8 feet is achieved by kicking the ball at an angle of 45.6 \degree.

Outcomes

MA.912.AR.3.4

Write a quadratic function to represent the relationship between two quantities from a graph, a written description or a table of values within a mathematical or real-world context.

MA.912.AR.3.8

Solve and graph mathematical and real-world problems that are modeled with quadratic functions. Interpret key features and determine constraints in terms of the context.

MA.912.DP.2.8

Fit a quadratic function to bivariate numerical data that suggests a quadratic association and interpret any intercepts or the vertex of the model. Use the model to solve real-world problems in terms of the context of the data.

MA.912.F.1.1

Given an equation or graph that defines a function, determine the function type. Given an input-output table, determine a function type that could represent it.

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