b. Make a table of values that includes the
vertex.
c. Use this information to graph the function.
Study Guide and Review - Chapter 4
State whether each sentence is true or false . If
false , replace the underlined term to make a
true sentence.
10. SOLUTION: a. To find the y-intercept, substitute x = 0 in the
function.
The y-intercept is 15.
2. The graph of a quadratic function is called a
parabola.
SOLUTION: true
The x-coordinate of the vertex is given by
.
So:
4. The axis of symmetry will intersect a parabola in one
point called the vertex.
The line of symmetry passes through the vertex.
Therefore, the equation of the axis of symmetry is
SOLUTION: true
.
b.
6. The equation
is known as the discriminant.
SOLUTION: False, Quadratic Formula
c. The graph of the quadratic function is:
8. The two numbers 2 + 3i and 2 – 3i are called
complex conjugates.
SOLUTION: true
Complete parts a–c for each quadratic function.
a. Find the y-intercept, the equation of the axis
of symmetry, and the x-coordinate of the
vertex.
b. Make a table of values that includes the
vertex.
c. Use this information to graph the function.
12. SOLUTION: a.
To find the y-intercept, substitute x = 0 in the
function.
The y-intercept is –1.
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SOLUTION: a.
The x-coordinate of the vertex is given by
Page. 1
Study
Guide and Review - Chapter 4
Determine whether each function has a
maximum or minimum value and find the
maximum or minimum value. Then state the
domain and range of the function.
12. SOLUTION: a.
To find the y-intercept, substitute x = 0 in the
function.
The y-intercept is –1.
14. The x-coordinate of the vertex is given by
.
SOLUTION: The sign of a is negative so the function has a
maximum value.
Equation of the axis of the symmetry is x = 2.
b.
The maximum value occurs at the vertex. Substitute
in .
c. The graph of the quadratic function is:
Determine whether each function has a
maximum or minimum value and find the
maximum or minimum value. Then state the
domain and range of the function.
So, the maximum value is
14. SOLUTION: The sign of a is negative so the function has a
maximum value.
.
Domain = {all real numbers}
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Solve each equation by graphing. If exact roots
cannot be found, state the consecutive integers
between which the roots are located.
.
Domain = {all real numbers}
The graph intersects the x-axis between –2 and –1,
and between 2 and 3. So the roots of the equation
are between –2 and –1 and between 2 and 3.
Study Guide and Review - Chapter 4
Write a quadratic equation in standard form with
the given roots.
Solve each equation by graphing. If exact roots
cannot be found, state the consecutive integers
between which the roots are located.
20. 5, 6
16. SOLUTION: SOLUTION: 2
Graph the function y = x – x – 20.
22. –4, 2
SOLUTION: The graph intersects the x-axis at –4 and 5.
So, the roots of the equation are x = –4 and x = 5.
24. 18. SOLUTION: SOLUTION: Graph the function.
Solve each equation by factoring.
26. SOLUTION: The graph intersects the x-axis between –2 and –1,
and between 2 and 3. So the roots of the equation
are between –2 and –1 and between 2 and 3.
Write a quadratic equation in standard form with
the given roots.
20. 5, 6
The solution set is {–3, 4}.
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SOLUTION: 28. Study Guide and Review - Chapter 4
Solve each equation by factoring.
32. (6 + 2i) – (4 – 3i)
SOLUTION: Subtract the real and the imaginary parts separately.
SOLUTION: 26. The solution set is {–3, 4}.
34. ELECTRICITY The impedance in one part of a
series circuit is 3 + 2j ohms, and the impedance in
the other part of the circuit is 4 – 3j ohms. Add these
complex numbers to find the total impedance in the
circuit.
SOLUTION: Add the complex numbers.
28. SOLUTION: The total impedance in the circuit is 7 – j ohms.
Solve each equation.
36. The solution set is
SOLUTION: .
Simplify.
30. 38. SOLUTION: SOLUTION: 32. (6 + 2i) – (4 – 3i)
Find the value of c that makes each trinomial a
perfect square. Then write the trinomial as a
Page 4
perfect square.
SOLUTION: Subtract
the
real andbythe
imaginary parts separately.
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40. Study Guide and Review - Chapter 4
Find the value of c that makes each trinomial a
perfect square. Then write the trinomial as a
perfect square.
44. SOLUTION: To find the value of c, divide the coefficient of x by 2
and square it.
40. SOLUTION: To find the value of c, divide the coefficient of x by
2, and square it.
Substitute the value of c in the trinomial.
Substitute the value of c in the trinomial.
Solve each equation by completing the square.
42. 46. SOLUTION: To find the value of c, divide the coefficient of x by 2
and square it.
SOLUTION: Substitute the value of c in the trinomial.
The solution set is {–1, 7}.
48. 44. SOLUTION: SOLUTION: To find the value of c, divide the coefficient of x by 2
and square it.
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The solution set is
The solution set is {–1, 7}.
Study
Guide and Review - Chapter 4
.
50. FLOOR PLAN Mario’s living room has a length 6
feet wider than the width. The area of the living
room is 280 square feet. What are the dimensions of
his living room?
48. SOLUTION: SOLUTION: Let x feet be the width of the room.
So, the length is x + 6 feet.
The area of a rectangle is given by
l is length and w is the width.
, where
Therefore:
x cannot be negative.
So, x = 14.
The solution set is
.
Therefore, the width of the room is 14 feet, and the
length is 20 feet.
50. FLOOR PLAN Mario’s living room has a length 6
feet wider than the width. The area of the living
room is 280 square feet. What are the dimensions of
his living room?
Complete parts a–c for each quadratic equation.
a. Find the value of the discriminant.
b. Describe the number and type of roots.
SOLUTION: Let x feet be the width of the room.
c. Find the exact solutions by using the
Quadratic Formula.
So, the length is x + 6 feet.
The area of a rectangle is given by
l is length and w is the width.
, where
Therefore:
52. SOLUTION: a.
Substitute the values of a, b, and c.
x cannot be negative.
So, x = 14.
b. Since the discriminant is a perfect square, the
quadratic equation has 2 rational real roots.
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Therefore, the width of the room is 14 feet, and the
length is 20 feet.
Page 6
c.
Substitute the values of a, b, and c.
So, x = 14.
Therefore, the width of the room is 14 feet, and the
length
is 20
feet.
Study
Guide
and
Review - Chapter 4
The solution set is {–8, 4}.
Complete parts a–c for each quadratic equation.
54. a. Find the value of the discriminant.
SOLUTION: b. Describe the number and type of roots.
a.
Substitute the values of a, b, and c.
c. Find the exact solutions by using the
Quadratic Formula.
b. Since the discriminant is a perfect square, the
quadratic equation has 2 rational real roots.
52. SOLUTION: c.
a.
Substitute the values of a, b, and c.
Substitute the values of a, b, and c.
b. Since the discriminant is a perfect square, the
quadratic equation has 2 rational real roots.
c.
Substitute the values of a, b, and c.
The solution set is
.
56. SOLUTION: The solution set is {–8, 4}.
a.
Substitute the values of a, b, and c.
b. Since the discriminant is zero, the quadratic
equation has 1 real rational root.
54. SOLUTION: a.
Substitute the values of a, b, and c.
c.
b. Since the discriminant is a perfect square, the
quadratic equation has 2 rational real roots.
Substitute the values of a, b, and c.
c.
Substitute the values of a, b, and c.
The solution set is
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.
Page 7
58. PHYSICAL SCIENCE Lauren throws a ball with
Study
Guide
andset
Review
- Chapter
4
The
solution
is
.
56. SOLUTION: a.
Substitute the values of a, b, and c.
The solution set is
.
58. PHYSICAL SCIENCE Lauren throws a ball with
an initial velocity of 40 feet per second. The equation
2
for the height of the ball is h = –16t + 40t + 5,
where h represents the height in feet and t
represents the time in seconds. When will the ball hit
the ground?
SOLUTION: b. Since the discriminant is zero, the quadratic
equation has 1 real rational root.
Substitute h = 0 in the equation and solve for t.
c.
Substitute the values of a, b, and c.
The solution set is
.
Since t cannot be negative, t = 2.62.
Therefore, it takes about 2.62 seconds to hit the
ground.
58. PHYSICAL SCIENCE Lauren throws a ball with
an initial velocity of 40 feet per second. The equation
2
for the height of the ball is h = –16t + 40t + 5,
where h represents the height in feet and t
represents the time in seconds. When will the ball hit
the ground?
Write each quadratic function in vertex form, if
not already in that form. Then identify the
vertex, axis of symmetry, and direction of
opening. Then graph the function.
SOLUTION: 60. Substitute h = 0 in the equation and solve for t.
SOLUTION: The vertex is at (–3, –26).
The line of symmetry is x = –3.
Since t cannot be negative, t = 2.62.
Therefore, it takes about 2.62 seconds to hit the
ground.
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Write each quadratic function in vertex form, if
2
Here, a, the coefficient of x , is greater than zero.
So, the graph opens up.
Page 8
Since t cannot be negative, t = 2.62.
Therefore, it takes about 2.62 seconds to hit the
ground.
Study
Guide and Review - Chapter 4
Write each quadratic function in vertex form, if
not already in that form. Then identify the
vertex, axis of symmetry, and direction of
opening. Then graph the function.
62. SOLUTION: 60. SOLUTION: The vertex is at (–6, –83).
The line of symmetry is x = –6.
2
Here, a, the coefficient of x , is greater than zero.
So, the graph opens up.
The vertex is at (–3, –26).
The line of symmetry is x = –3.
2
Here, a, the coefficient of x , is greater than zero.
So, the graph opens up.
Graph each quadratic inequality.
64. SOLUTION: 62. Graph the related function
Since the inequality symbol is
be solid.
Test the point (–2, 2).
SOLUTION: The vertex is at (–6, –83).
The line of symmetry is x = –6.
. , the parabola should
Shade the region that contains (–2, 2).
2
Here, a, the coefficient of x , is greater than zero.
So, the graph opens up.
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Study Guide and Review - Chapter 4
Graph each quadratic inequality.
66. 64. SOLUTION: SOLUTION: Graph the related function
Since the inequality symbol is
be solid.
Test the point (–2, 2).
Graph the related function
. , the parabola should
.
Since the inequality symbol is >, the parabola should
be dashed
Test the point (3, 2).
Shade the region that contains (–2, 2).
Shade the region that contains the point (3, 2).
66. SOLUTION: Graph the related function
.
68. Solomon wants to put a deck along two sides of his
garden. The deck width will be the same on both
sides and the total area of the garden and deck
cannot exceed 500 square feet. How wide can the
deck be?
Since the inequality symbol is >, the parabola should
be dashed
Test the point (3, 2).
SOLUTION: The area of a rectangle is given by
l is the length and w is the width.
Shade the region that contains the point (3, 2).
So:
, where
Solve the inequality.
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Page 10
inequality is
.
Therefore, the width of the deck should be between
0 and 5 feet.
Study Guide and Review - Chapter 4
68. Solomon wants to put a deck along two sides of his
garden. The deck width will be the same on both
sides and the total area of the garden and deck
cannot exceed 500 square feet. How wide can the
deck be?
Solve each inequality using a graph or
algebraically.
70. SOLUTION: SOLUTION: The area of a rectangle is given by
l is the length and w is the width.
The inequality is true for all x.
So, the solution is the set of real numbers.
, where
So:
72. SOLUTION: Solve the inequality.
Solve the related equation
Since x should be positive, the solution set of the
inequality is
.
.
Test one value of x less than –1.69, one between –
1.69 and 0.44, and one greater than 0.44.
Test the point –2.
Therefore, the width of the deck should be between
0 and 5 feet.
Solve each inequality using a graph or
algebraically.
Test the point 0.
70. SOLUTION: Test the point 1.
Therefore, the solution set is
The inequality is true for all x.
So, the solution is the set of real numbers.
.
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72. Page 11