Topics
1. Analyzing Functions
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United States of AmericaTennessee
Algebra 2

1.04 Comparing functions across representations

Lesson
Practice
Lesson

Concept summary

Functions can be represented in a variety of ways, including equations, tables, and graphs. It is important to be able to compare functions whether they are represented in similar or different ways.

Useful information can usually be obtained by comparing key features of the functions. Remember that key features include:

Domain

The complete set of possible values of the input of a function or relation. The domain may vary depending on the context.

A curved function drawn on a coordinate plane with an unfilled endpoint at the left most part of the function and a filled endpoint at the right most part of the function. A segment across the top of the function marks the domain with an unfilled enpoint to the left and a filled endpoint to the right
Range

The complete set of possible values of the output of a relation or function

A curved function drawn on a coordinate plane with an unfilled endpoint at the left most part of the function and a filled endpoint at the right most part of the function. A vertical segment to the right of the function marks the range with filled endpoints at the top and bottom
x- and y-intercepts

Points where a function intersects with an axis. An x-intercept occurs where y = 0 and a y-intercept occurs where x = 0.

The line y=-x+3 drawn in a coordinate plane. The point (3,0) is labeled x-intercept
Vertex

The maximum or minimum point of a function, written as an ordered pair. Not every function type has a vertex.

The graph of a quadratic function that opens upward with a point plotted at its minimum labeled vertex
Rate of change

The ratio of change between output values to the corresponding change in input values. For lines and line segments, this is equivalent to slope.

Increasing/decreasing interval

A connected region of the domain in which the output values become higher (increase) or lower (decrease) as the input values become larger.

Positive/negative interval

A connected region of the domain in which the output values all lie above the x-axis (positive) or all lie below the x-axis (negative).

End behaviour

The trend of the graph of a function on either side; the y-value that a function obtains or approaches as the x-values get far from zero.

Asymptote

A line which the graph of a function approaches as one or both variables tend towards \pm \infty.

A decreasing exponential function approaching but never touching a dashed horizontal line labeled asymptote

Worked examples

Example 1

Consider the functions shown below.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
a

Compare the number and location of the x-intercepts of the two functions.

Approach

We can label the x-intercepts on each graph to more clearly see where they are.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y

Solution

Looking at the graphs, we can see that the two functions have the same two x-intercepts, at the points \left(-1, 0\right) and \left(3, 0\right).

b

State which function has a lower minimum value.

Approach

The minimum value of each function will be the y-value of its vertex. We can label the vertex on each graph to more clearly see where they are.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y

Solution

Looking at the graphs, we can see that the vertex of the parabola is at \left(1, -4\right) while the vertex of the absolute value function is at \left(1,- 2\right).

So the minimum value of the quadratic function is -4, which is lower than the minimum value of the absolute value function which is -2.

c

Determine the domain over which the absolute value function is higher than the quadratic function.

Approach

In part (a) we determined that the two functions intersect at their two x-intercepts. We can use this to split the domain of the functions into three regions:

  • The region where x < -1, to the left of \left(-1, 0\right).
  • The region where -1 < x < 3, in between \left(-1, 0\right) and \left(3, 0\right).
  • The region where x > 3, to the right of \left(3, 0\right).

It may also be useful to sketch a graph of each function on the same coordinate plane.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y

Solution

We know from part (b) that the vertex of the absolute value function is higher than the vertex of the parabola. Looking at both graphs on the same plane, we can see that the absolute value function is higher than the quadratic function for -1 < x < 3.

d

Compare and contrast the end behavior of the two functions.

Solution

The output values of both functions continue to increase as the value of x gets further away from zero, on both sides. That is,f\left(x\right) \to \infty \text{ as } x \to \pm \inftyis true for both functions.

The quadratic function, however, increases at an increasing rate, while the slope of the absolute value function changes at a constant rate. This means the quadratic function tends towards infinity faster than the absolute value function.

Reflection

Remember that when we look at increasing or decreasing intervals for functions, we usually consider how the function values change as we move from left to right.

The exception to this is when we are looking at the end behavior of functions, in which case we consider how the function values change as the input values get further from zero instead.

Example 2

Consider the functions shown below. Assume that the domain of g is all real numbers.

  • f\left(x\right) = 2^x - 1
  • x-3-2-10123
    g\left(x\right)-4-202468
a

Determine which function has a higher y-intercept.

Approach

Remember that the y-intercept of a function occurs when x = 0. We can use this to evaluate the y-intercept of f and identify the y-intercept of g.

Solution

For f, we have

\displaystyle f\left(x\right)\displaystyle =\displaystyle 2^x - 1
\displaystyle f\left(0\right)\displaystyle =\displaystyle 2^0 - 1
\displaystyle =\displaystyle 1 - 1
\displaystyle =\displaystyle 0

For g, we can see from the table that g\left(0\right) = 2.

So the y-intercept of f is the point \left(0, 0\right) and the y-intercept of g is the point \left(0, 2\right), and therefore g has a higher y-intercept.

b

Determine which function has a horizontal asymptote.

Solution

The equation for function f has a variable in the exponent, so f is an exponential function. This means it has a horizontal asymptote.

Function g, on the other hand, appears to be a linear function from the values shown in the table, and so it does not have a horizontal asymptote.

c

Determine the negative interval(s) for each function.

Approach

To determine negative intervals for these functions, we want to first find the location of any x-intercept, since these are the points where the function can change from being negative to positive or vice-versa.

Solution

For f, we know from part (a) that it has an x-intercept at the origin, \left(0, 0\right). Since it is an exponential function, this will also be its only x-intercept.

Checking points on either side, we can see thatf\left(-1\right) = 2^{-1} - 1 = -\dfrac{1}{2} < 0whilef\left(1\right) = 2^1 - 1 = 1 > 0 So the negative interval for f is to the left of the origin: \left(-\infty, 0\right).

For g, we can see that it has an x-intercept at \left(-1, 0\right), and as a linear function it has only one. We can also see that the function values are negative to the left of the x-intercept and positive to the right of it.

So the negative interval for g is \left(-\infty, -1\right).

d

Compare the range of each function over their negative intervals.

Solution

Function f is an exponential function with an asymptote. In particular, the value of 2^x is always greater than 0, and so the value of 2^x - 1 is always greater than -1.

So, over its negative interval f is bounded between -1 and 0. That is, the range of f over its negative interval is \left(-1, 0\right).

Function g, as a linear function, however, has no lowest value that it will obtain. So its range over its negative interval is \left(-\infty, 0\right).

Outcomes

A2.F.IF.B.6

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

A2.F.IF.B.6.A

Compare properties of two different functions. Functions may be of different types and/or represented in different ways.

A2.F.IF.B.6.B

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

A2.F.BF.B.3

Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs.

A2.MP1

Make sense of problems and persevere in solving them.

A2.MP2

Reason abstractly and quantitatively.

A2.MP3

Construct viable arguments and critique the reasoning of others.

A2.MP4

Model with mathematics.

A2.MP6

Attend to precision.

A2.MP7

Look for and make use of structure.

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