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4.06 Characteristics of linear functions

Lesson

Concept summary

The characteristics, or key features of a function include its:

  • domain and range
  • x- and y-intercepts
  • maximum or minimum value(s)
  • rate of change over specific intervals

Key features of a function are useful in helping to sketch the function, as well as to interpret information about the function in a given context.

In the case of linear functions, they will either be constant everywhere (a horizontal line), increasing everywhere, or decreasing everywhere, due to their constant rate of change.

This also means that they will never have a maximum or minimum turning point, and will have at most one x-intercept (except for the horizontal line y = 0).

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An increasing linear function. It has an \\x-intercept at \left(-2, 0\right) and a y-intercept at \left(0, 2\right).
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A decreasing linear function. It has an \\x-intercept at \left(2, 0\right) and a y-intercept at \left(0, 1\right).

Worked examples

Example 1

Consider the function f\left(x\right) = 4x - 8.

a

Find the value of f\left(0\right).

Approach

To find the value of a function at a particular x-value, we simply substitute that value into the function.

Solution

We have \begin{aligned} f\left(0\right) & = 4\left(0\right) - 8 \\ & = -8 \end{aligned}

Reflection

The y-intercept of a function is the point on the function where x = 0. So we know that this function has a y-intercept at \left(0, -8\right).

b

Find the value of x which gives a function value of 0.

Approach

This means that we want to solve the equation f\left(x\right) = 0 using the expression we have for f\left(x\right).

Solution

We have \begin{aligned} f\left(x\right) & = 0 \\ 4x - 8 & = 0 \\ 4x & = 8 \\ x & = 2 \end{aligned}

Reflection

The x-intercepts of a function are the points on the function where y = 0. So we know that this function has a single x-intercept at \left(2, 0\right).

c

Sketch a graph of the function and label each intercept.

Approach

We found the two intercepts in parts (a) and (b). Since two points determine a line, we can connect the intercepts to create the graph of the function.

Solution

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Example 2

Whitney is traveling across the city by taking an Uber ride. The cost of the ride, in dollars, is given by C\left(x\right) = 2.4x + 2.8 where x is the distance traveled in miles. The minimum charge for a ride is \$10.

a

State the range of the function.

Solution

Since x is the distance traveled in miles, we know that x > 0, which corresponds to C\left(x\right) > 2.8. We are also told that the minimum charge is \$10.

Putting these together, we have that the range is \text{Range: } \left\{y\, \vert\, y \geq 10 \right\}

b

Determine the rate of change of the function, and state what it represents in context.

Approach

The rate of change of a linear function in the form y = mx + b is m, the constant being multiplied by the variable.

Solution

In this case, the function is C\left(x\right) = 2.4x + 2.8, so the rate of change is 2.4.

Since the output of C\left(x\right) is a value in dollars, and x is a value in miles, the rate of change is 2.4 dollars per mile, and it represents the additional cost of the trip per each extra mile traveled.

c

If Whitney is travelling a distance of 4 miles, determine the cost of her trip.

Approach

We want to find the value of C\left(4\right), and compare it to the minimum cost. The larger of the two values will be the cost of Whitney's trip.

Solution

We have \begin{aligned} C\left(4\right) & = 2.4\left(4\right) + 2.8 \\ & = 9.6 + 2.8 \\ & = 12.4 \end{aligned}

Since this is larger than the minimum amount, the cost of her trip will be \$12.40.

Example 3

Consider functions f and g shown in the graph. Compare the following features of each function:

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a

Domain

Approach

The domain of a function is the set of all x-values that correspond to a point on the graph.

Solution

As these are linear functions, with no domain restrictions from context, all real values of x correspond to a point on each line.

So the domain of both functions is all real values of x.

b

Range

Approach

The range of a function is the set of all y-values that correspond to a point on the graph.

Solution

We can see that all values of y correspond to a point on the graph of f, so its range will be all real values of y.

On the other hand, g is a constant function and so it only takes one y-value - in this case, 2. So the range of g is just y = 2.

c

Intercepts

Approach

The x-intercepts of a function are the points where it intersects the x-axis. Similarly, the y-intercept of a function is the point where it intersects the y-axis.

As these are linear functions, they will have at most one of each type of intercept.

Solution

We can see that function f crosses the x-axis at the point \left(2, 0\right), and crosses the y-axis at the point \left(0, -2\right).

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Function g crosses the y-axis at the point \left(0, 2\right). Since it is a constant function, it is parallel to the x-axis and so it does not have an x-intercept.

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d

Slope

Approach

The slope of a line is the ratio of its change in the vertical direction to the change in the horizontal direction.

That is, the slope is a measure of how far up (or down) the y-values change for each unit change in the x-values.

Solution

For function f we can see that at any point on the graph, if we move 1 unit to the right (in the x-direction), we also move 1 unit up (in the y-direction).

So the slope of f is 1.

Function g is a constant function, which has no change in its y-values. That is, at any point on the graph, moving 1 unit to the right corresponds to 0 units of change in the vertical direction.

So the slope of g is 0.

Outcomes

A1.N.Q.A.1

Use units as a way to understand real-world problems.*

A1.N.Q.A.1.B

Use appropriate quantities in formulas, converting units as necessary.*

A1.A.REI.D.5

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

A1.F.IF.B.4

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.*

A1.F.IF.B.5

Relate the domain of a function to its graph and, where applicable, to the context of the function it models. *

A1.F.IF.B.6

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate and interpret the rate of change from a graph.*

A1.F.IF.C.7

Graph functions expressed algebraically and show key features of the graph by hand and using technology.*

A1.F.IF.C.9

Compare properties of functions represented algebraically, graphically, numerically in tables, or by verbal descriptions.*

A1.F.IF.C.9.B

Compare properties of the same function on two different intervals or represented in two different ways.

A1.F.LE.B.3

Interpret the parameters in a linear or exponential function in terms of a context.*

A1.MP1

Make sense of problems and persevere in solving them.

A1.MP2

Reason abstractly and quantitatively.

A1.MP3

Construct viable arguments and critique the reasoning of others.

A1.MP4

Model with mathematics.

A1.MP6

Attend to precision.

A1.MP7

Look for and make use of structure.

A1.MP8

Look for and express regularity in repeated reasoning.

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