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The transformations of the graph of a quadratic function, including translations, reflections, and vertical compressions. It also includes instructions on how to find the function that best fits given data and how to determine the number of car club members after a certain number of years.
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Use a table of values to graph the following functions. State the domain and range.
y = x
2
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 (í1, 4). So the range is all real numbers greater than or equal to the minimum value of 4, or R = { y | y `
x y
í 3
í 2 5
í 1 4
y = 2 x
2
í 3 x + 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.
x y
í 2 15
í 1 6
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Practice Test - Chapter 9
The function is a parabola, so the domain is all real numbers. The graph opens upwards, so the vertex is a minimum
located at (í1, 4). So the range is all real numbers greater than or equal to the minimum value of 4, or R = { y | y `
y = 2 x
2
í 3 x + 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 í0.125}
x y
í 2 15
í 1 6
Consider y = x
2
í 7 x + 6.
Determine whether the function has a maximum or minimum value.
For y = x
2
± 7 x + 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 = x
2
± 7 x + 6, use a = 1, b = ±7, and c = 6.
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Practice Test - Chapter 9
The minimum of the function is í6.25. Therefore, R = { y | y í6.25} 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.
x
2
Graph the related function f ( x ) = x
2
The x - intercepts of the graph appear to be at ±5 and ±2, so the solutions are ±5 and ±2.
Check:
and
x
2
í 5 = í 3 x
Write the equation in standard form.
Graph the related function f ( x ) = x
2
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.
x ±4.5 ±4.4 ±4.3 ±4.2 ±4.
y 1.75 1.16 0.59 0.04 ±0.
x 1.1 1.2 1.3 1.4 1.
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Practice Test - Chapter 9
x
2
í 5 = í 3 x
Write the equation in standard form.
Graph the related function f ( x ) = x
2
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.
y 1.75 1.16 0.59 0.04 ±0.
x 1.1 1.2 1.3 1.4 1.
y ±0.49 0.04 0.59 1.16 1.
Describe how the graph of each function is related to the graph of f ( x ) = x
2
g ( x ) = x
2
í 5
The graph of f ( x ) = x
2
< 0, the graph of f ( x ) = x
2
is translated XQLWVGRZQ7KHUHIRUHWKHJUDSKRI y = x
2
± 5 is a translation of the graph
of y = x
2
shifted down 5 units.
g ( x ) = í 3 x
2
7KHJUDSKRI f ( x ) = ax
2
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 = í 3 x
2
is the graph of y = x
2
reflected across the x - axis and vertically expanded.
The function can be written f ( x ) = ax
2
x
2
2
vertically compressed and shifted up 4 units.
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Practice Test - Chapter 9
x
2
í x í 6 = 0
The solutions are 3 and ±2.
2 x
2
í 36 = í 6 x
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.
x
2
í x í 30 = 0
For this equation, a = 1, b = ±1, and c = ±30.
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Practice Test - Chapter 9
The solutions are 3 and ±6.
Solve each equation by using the Quadratic Formula. Round to the nearest tenth if necessary.
x
2
í x í 30 = 0
For this equation, a = 1, b = ±1, and c = ±30.
The solutions are 6 and ±5.
x
2
í 10 x = í 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.
2 x
2
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Practice Test - Chapter 9
The solutions are 2.5 and ±3.
Elias hits a baseball into the air. The equation h = ± 16 t
2
ball after t seconds. How long is the ball in the air?
The ball is in the air for about 3.8 seconds.
Graph {(í2, 4), (í1, 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.
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Practice Test - Chapter 9
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.
The data describes the linear function f ( x ) = 2 x + 1.
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:
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Practice Test - Chapter 9
Graph each function.
f ( x ) = | x í 1|
Make a table of values.
x f ( x )
f ( x ) = í| 2 x |
The negative in front of the absolute value makes the graph point down. Make a table of values.
x f ( x )
f ( x ) =
Make a table of values.
x f ( x )
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Practice Test - Chapter 9
f ( x ) =
Make a table of values.
x f ( x )
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 )
Determine the domain and range of the function graphed below.
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Practice Test - Chapter 9