Use a table of values to graph the following functions. State the domain and range.
y = x2 + 2x + 5
Graph the ordered pairs, and connect them to create a smooth curve. The parabola extends to infinity.
The function is a parabola, so the domain is all real numbers. The graph opens upwards, so the vertex is a minimum
located at (1, 4). So the range is all real numbers greater than or equal to the minimum value of 4, or R = {y| y
x y
3 8
2 5
1 4
0 5
1 8
2 13
y = 2x2 3x + 1
Graph the ordered pairs, and connect them to create a smooth curve. The parabola extends to infinity.
The function is a parabola, so the domain is all real numbers. The range is all real numbers greater than or equal to
the minimum value.
R = {y | y 0.125}
x y
2 15
1 6
0 1
1 0
2 3
3 10
Consider y = x2 7x + 6.
Determine whether the function has a maximum or minimum value.
For y = x2 7x + 6, a = 1, b = 7, and c = 6. Because a is positive, the graph opens upward, so the function has a
minimum value.
State the maximum or minimum value.
Method 1: Find the vertex of the parabola.
First, find the x-coordinate of the vertex. For the function y = x2 7x + 6, use a = 1, b = 7, and c = 6.
Substitute 3.5 for x in the function to find the value of the y-
So, the minimum value is 6.25.
Method 2: Use a graphing calculator to graph the related function f(x) = x2 7x + 6. Use the minimum option on
the 2nd [CALC] menu to determine the minimum value.
Therefore, the minimum value is 6.25.
What are the domain and range?
Any number can be substituted for x in the function. So, t
The minimum of the function is 6.25. Therefore, R = {y | y 6.25} because the range is all real numbers that are
Solve each equation by graphing. If integral roots cannot be found, estimate the roots to the nearest
tenth.
x2 + 7x + 10 = 0
Graph the related function f(x) = x2 + 7x + 10.
The x-intercepts of the graph appear to be at 5 and 2, so the solutions are 5 and 2.
Check:
and
x2 5 = 3x
Write the equation in standard form.
Graph the related function f(x) = x2 + 3x 5.
The x-intercepts are located between 5 and 4 and between 1 and 2. Make a table using an increment of 0.1 for
the x-values located between 5 and 4 and between 1 and 2.
For each table, the function value that is closest to zero when the sign changes is 0.04. Thus, the roots are
approximately 4.2 and 1.2.
x 4.5 4.4 4.3 4.2 4.1
y 1.75 1.16 0.59 0.04 0.49
x 1.1 1.2 1.3 1.4 1.5
y 0.49 0.04 0.59 1.16 1.75
Describe how the graph of each function is related to the graph of f(x) = x2.
g(x) = x2 5
The graph of f(x) = x2 + c represents a vertical translation of the parent graph. The value of c is 5, and 5 < 0. If c
< 0, the graph of f(x) = x2 is translated y = x2 5 is a translation of the graph
of y = x2 shifted down 5 units.
g(x) = 3x2
f(x) = ax2 expands or compresses the parent graph vertically. Since |a| > 1, the graph is vertically
expanded. Since a < 1, the graph is reflected across the x-axis. So, the graph of y = 3x2 is the graph of y = x2
reflected across the x-axis and vertically expanded.
The function can be written f(x) = ax2 + c, where a = c = 4. Since 4 > 0 and 0 < y =
x2 + 4 is the graph of y = x2 vertically compressed and shifted up 4 units.
Which is an equation for the function shown in the graph?
y = 3x2
y = 3x2 + 1
y = x2 + 2
y = 3x2 + 2
The parabola opens downward, so the a in the equation must be negative. Therefore, choices B and C can be
eliminated. The y-intercept is at the point (0, c), or in this case (0, 2). Therefore c = 2, so choice A can be eliminated.
So, the correct choice is choice D.
Solve each equation by completing the square.
x2 + 2x + 5 = 0
No real number has a negative square. So, this equation has no real solutions.
x2 x 6 = 0
The solutions are 3 and 2.
2x2 36 = 6x
Rewrite the equation in standard form.
The solutions are 3 and 6.
Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary.
x2 x 30 = 0
For this equation, a = 1, b = 1, and c = 30.
The solutions are 6 and 5.
x2 10x = 15
Rewrite the equation in standard form.
For this equation, a = 1, b = 10, and c = 15.
The solutions are 1.8 and 8.2.
2x2 + x 15 = 0
For this equation, a = 2, b = 1, and c = 15.
The solutions are 2.5 and 3.
Elias hits a baseball into the air. The equation h = 16t2 + 60t +3 models the height h in feet of the
ball after t seconds. How long is the ball in the air?
The ball is in the air for about 3.8 seconds.
Graph {(2, 4), (1, 1), (0, 0), (1, 1), (2, 4)}. Determine whether the ordered pairs represent a linear function, a
quadratic function, or an exponential function.
The points are reflected across a line of symmetry. They represent a quadratic function.
Look for a pattern in the table to determine which kind of model best describes the data.
Calculate the first differences.
Find the slope and y-
Use the slope and one point from the table to find the y-
The data describes the linear function f(x) = 2x + 1.
CAR CLUB The table shows the number of car club members for four consecutive years after it began.
a. Determine which model best represents the data.
b. Write a function that models the data.
c. Predict the number of car club members after 6 years.
a. First compare the first differences in the number of members:
Since they're not equal, compare the second differences:
Since the second differences are not equal, look for a common ratio:
b. A function that fits the data should be an exponential function with a ratio of 2. We are looking for a function of
the form . Use a point from the data given to determine the value of a.
The function that best fits the data is:
c. To find the number of car club members after 6 years, use the function found in part b and plug in 6 for x.
Graph each function.
f(x) = |x 1|
Make a table of values.
x f(x)
3 4
2 3
1 2
0 1
1 0
2 1
3 2
4 3
f(x) = |2x|
The negative in front of the absolute value makes the graph point down. Make a table of values.
x f(x)
2 4
1 2
0 0
1 2
2 4
f(x) =
Make a table of values.
x f(x)
0 0
0.5 0
1 1
1.5 1
2 2
2.5 2
3 3
f(x) =
This is a piecewise-defined function. Make a table of values. Be sure to include the domain values for which the
function changes.
x f(x)
2 5
1 3
0 1
1 1
2 1
3 0
4 1
Determine the domain and range of the function graphed below.
The graph will cover all possible values of x, so the domain is all real numbers. The graph will go no lower than y =
2, so the domain is all real numbers and the range is {y | y 2}.
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Practice Test - Chapter 9
