State whether each sentence is true or false . If false ...

Study Guide and Review - Chapter 9

State whether each sentence is true orfalse . Iffalse , replace the underlined term to make a true sentence.

1. The axis of symmetry of a quadratic function can be found by using the equation x =

.

SOLUTION: The shape of the graph of a quadratic function is called a parabola. Parabolas are symmetric about a central line called the axis of symmetry. The axis of symmetry of a quadratic function can be found by using the equation

. The statement is true.

2. The vertex is the maximum or minimum point of a parabola.

SOLUTION: The axis of symmetry intersects a parabola at only one point, called the vertex. The lowest point on the graph is the minimum, and the highest point on the graph is the maximum. The vertex is the maximum or minimum point of a parabola. The statement is true.

3. The graph of a quadratic function is a straight line.

SOLUTION: Quadratic functions are nonlinear and can be written in the form f (x) = ax2 + bx + c, where a 0. So, the statement is false. The graph of a quadratic function is a parabola .

4. The graph of a quadratic function has a maximum if the coefficient of the x2-term is positive.

SOLUTION: When a > 0, the graph of y = ax2 + bx + c opens upward. The lowest point on the graph is the minimum. So, the statement is false. The graph of a quadratic function has a minimum if the coefficient of the x2-term is positive.

5. A quadratic equation with a graph that has two x-intercepts has one real root.

SOLUTION: The solutions or roots of a quadratic equation can be identified by finding the x-intercepts of the related graph. So, the statement is false. A quadratic equation with a graph that has two x-intercepts has two real roots.

6. The expression b2 - 4ac is called the discriminant.

SOLUTION: In the Quadratic Formula, the expression under the radical sign, b2 - 4ac, is called the discriminant. The statement is true.

7. The solutions of a quadratic equation are called roots.

SOLUTION: The solutions of a quadratic equation are called roots. The statement is true.

8. The graph of the parent function is translated down to form the graph of f (x) = x2 + 5.

SOLUTION:

If c > 0 in f (x) = x2 + c, the graph of the parent function is translated units up. So, the statement is false. The

eSolutgiornaspMhaonfuathl -ePpoawreernedt fbuynCcotginoenrois translated up 5 units to form the graph of f (x) = x2 + 5.

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State whether each sentence is true orfalse . Iffalse , replace the underlined term to make a true sentence.

true.

7. The solutions of a quadratic equation are called roots.

StudSyOGLuUidTeIOanNd: Review - Chapter 9 The solutions of a quadratic equation are called roots. The statement is true.

8. The graph of the parent function is translated down to form the graph of f (x) = x2 + 5.

SOLUTION: If c > 0 in f (x) = x2 + c, the graph of the parent function is translated units up. So, the statement is false. The graph of the parent function is translated up 5 units to form the graph of f (x) = x2 + 5.

State whether each sentence is true orfalse . Iffalse , replace the underlined term to make a true sentence. 9. The range of the greatest integer function is the set of all real numbers.

SOLUTION: The statement is false. The domain of the greatest integer function is the set of all real numbers. The range is all integers.

State whether each sentence is true orfalse . Iffalse , replace the underlined term to make a true sentence. 10. A function that is defined differently for different parts of its domain is called a piecewise-defined function.

SOLUTION: true

Consider each equation. a. Determine whether the function has a maximum or minimum value.

b. State the maximum or minimum value. c. What are the domain and range of the function? 11. y = x2 - 4x + 4

SOLUTION: a. For y = x2 ? 4x + 4, a = 1, b = ?4, and c = 4. Because a is positive, the graph opens upward, so the function has a minimum value.

b. The minimum value is the y -coordinate of the vertex. The x-coordinate of the vertex is

.

The x-coordinate of the vertex is x = 2. Substitute this value into the function to find they -coordinate.

The minimum value is 0. c. The domain is all real numbers. The range is all real numbers greater than or equal to the minimum value, or {y|y 0}.

12. y = -x2 + 3x

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a. For y = -x2 + 3x, a = ?1, b = 3, and c = 0. Because a is negative, the graph opens downward, so the function has

a maximum value.

The minimum value is 0. c. The domain is all real numbers. The range is all real numbers greater than or equal to the minimum value, or {y|y Study 0G}u. ide and Review - Chapter 9

12. y = -x2 + 3x

SOLUTION: a. For y = -x2 + 3x, a = ?1, b = 3, and c = 0. Because a is negative, the graph opens downward, so the function has a maximum value.

b. The maximum value is the y -coordinate of the vertex. The x-coordinate of the vertex is

.

The x-coordinate of the vertex is x = 1.5. Substitute this value into the function to find they -coordinate.

The maximum value is 2.25. c. The domain is all real numbers. The range is all real numbers less than or equal to the minimum value, or {y|y 2.25}.

13. y = x2 - 2x - 3

SOLUTION: a. For y = x2 - 2x - 3, a = 1, b = ?2, and c = ?3. Because a is positive, the graph opens upward, so the function has a minimum value.

b. The minimum value is the y -coordinate of the vertex. The x-coordinate of the vertex is

.

The x-coordinate of the vertex is x = 1. Substitute this value into the function to find they -coordinate.

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The minimum value is ?4.

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The maximum value is 2.25. Studcy. GThueiddeomanadinRiseavlilerwea-l Cnuhmapbetersr. 9The range is all real numbers less than or equal to the minimum value, or {y|y

2.25}.

13. y = x2 - 2x - 3

SOLUTION: a. For y = x2 - 2x - 3, a = 1, b = ?2, and c = ?3. Because a is positive, the graph opens upward, so the function has a minimum value.

b. The minimum value is the y -coordinate of the vertex. The x-coordinate of the vertex is

.

The x-coordinate of the vertex is x = 1. Substitute this value into the function to find they -coordinate.

The minimum value is ?4. c. The domain is all real numbers. The range is all real numbers greater than or equal to the minimum value, or {y|y ?4}.

14. y = -x2 + 2

SOLUTION: a. For y = -x2 + 2, a = ?1, b = 0, and c = 2. Because a is negative, the graph opens downward, so the function has a maximum value.

b. The maximum value is the y -coordinate of the vertex. The x-coordinate of the vertex is

.

The x-coordinate of the vertex is x = 0. Substitute this value into the function to find they -coordinate.

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The maximum value is 2.

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The minimum value is ?4. Studcy. GThueiddeomanadinRiseavlilerwea-l Cnuhmapbetersr. 9The range is all real numbers greater than or equal to the minimum value, or {y|y

?4}.

14. y = -x2 + 2

SOLUTION: a. For y = -x2 + 2, a = ?1, b = 0, and c = 2. Because a is negative, the graph opens downward, so the function has a maximum value.

b. The maximum value is the y -coordinate of the vertex. The x-coordinate of the vertex is

.

The x-coordinate of the vertex is x = 0. Substitute this value into the function to find they -coordinate.

The maximum value is 2. c. The domain is all real numbers. The range is all real numbers less than or equal to the maximum value, or {y|y 2}.

15. BASEBALL A toy rocket is launched with an upward velocity of 32 feet per second. The equation h = -16t2 + 32t gives the height of the rocket t seconds after it is launched. a. Determine whether the function has a maximum or minimum value. b. State the maximum or minimum value. c. State a reasonable domain and range of this situation.

SOLUTION: a. For h = -16t2 + 32t, a = ?16, b = 32, and c = 0. Because a is negative, the graph opens downward, so the function has a maximum value.

b. The maximum value is the y -coordinate of the vertex. The x-coordinate of the vertex is

.

The x-coordinate of the vertex is x = 1. Substitute this value into the function to find they -coordinate.

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The maximum value is 2. Studcy. GThueiddeomanadinRiseavlilerwea-l Cnuhmapbetersr. 9The range is all real numbers less than or equal to the maximum value, or {y|y

2}.

15. BASEBALL A toy rocket is launched with an upward velocity of 32 feet per second. The equation h = -16t2 + 32t gives the height of the rocket t seconds after it is launched. a. Determine whether the function has a maximum or minimum value. b. State the maximum or minimum value. c. State a reasonable domain and range of this situation.

SOLUTION: a. For h = -16t2 + 32t, a = ?16, b = 32, and c = 0. Because a is negative, the graph opens downward, so the function has a maximum value.

b. The maximum value is the y -coordinate of the vertex. The x-coordinate of the vertex is

.

The x-coordinate of the vertex is x = 1. Substitute this value into the function to find they -coordinate.

The maximum value is 16. c. The rocket will be in the air for a total of 2 seconds. It will go from the ground to a maximum height of 16 feet, and then it will return to the ground. Therefore, a reasonable domain for this situation is D = {t | 0 t 2} and a reasonable range is R = {h | 0 h 16}. Solve each equation by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth. 16. x2 - 3x - 4 = 0 SOLUTION: Graph the related function f (x) = x2 - 3x - 4.

The x-intercepts of the graph appear to be at ?1 and 4, so the solutions are ?1 and 4. eSolutCioHnsEMCaKnu:aCl -hPeocwkereeadcbhy sCoolgunteioron in the original equation.

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The maximum value is 16. c. The rocket will be in the air for a total of 2 seconds. It will go from the ground to a maximum height of 16 feet, StudaynGd uthiedne iatnwdilRl reevtuierwn t-oCthheagprtoeurn9d. Therefore, a reasonable domain for this situation is D = {t | 0 t 2} and a reasonable range is R = {h | 0 h 16}.

Solve each equation by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth. 16. x2 - 3x - 4 = 0

SOLUTION: Graph the related function f (x) = x2 - 3x - 4.

The x-intercepts of the graph appear to be at ?1 and 4, so the solutions are ?1 and 4. CHECK: Check each solution in the original equation.

Therefore, the solutions are ?1 and 4. 17. -x2 + 6x - 9 = 0

SOLUTION: Graph the related function f (x) = -x2 + 6x - 9.

The x-intercept of the graph appears to be only at 3, so the only solution is 3. CHECK: Check the solution in the original equation.

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StudTyhGerueifdoerea,nthdeRsoelvuiteiwon-sCarhea?p1tearn9d 4.

17. -x2 + 6x - 9 = 0 SOLUTION: Graph the related function f (x) = -x2 + 6x - 9.

The x-intercept of the graph appears to be only at 3, so the only solution is 3. CHECK: Check the solution in the original equation.

Therefore, the solution is 3. 18. x2 - x - 12 = 0

SOLUTION: Graph the related function f (x) = x2 - x - 12.

The x-intercepts of the graph appear to be at ?3 and 4, so the solutions are ?3 and 4. CHECK: Check each solution in the original equation.

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Therefore, the solutions are ?3 and 4.

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