Quadratic Functions And Equations Quiz 1

1. Find the zeros of

y equals x squared minus 8 x plus 12

from its graph below.

2 and 6

-6 and -2

0 and 4

-4 and 4

The graph of the given function intersects the x-axis at the points x=2 and x=6. These points are called the zeros of the function because they are the values of x for which the function equals zero. Therefore, the correct answer is 2 and 6.

Explanation

The graph of the given function intersects the x-axis at the points x=2 and x=6. These points are called the zeros of the function because they are the values of x for which the function equals zero. Therefore, the correct answer is 2 and 6.

2. The vertex of this parabola shows that the __________ value of the function is _____.

Minimum; -5

Maximum; 1

Minimum; -9

Maximum; 2

The vertex of a parabola represents the highest or lowest point on the graph, depending on whether the parabola opens upward or downward. In this case, since the vertex is at a negative value (-9), it indicates that the function has a minimum value. Therefore, the correct answer is "minimum; -9."

Explanation

The vertex of a parabola represents the highest or lowest point on the graph, depending on whether the parabola opens upward or downward. In this case, since the vertex is at a negative value (-9), it indicates that the function has a minimum value. Therefore, the correct answer is "minimum; -9."

3. If you graph

y equals x squared minus 4 x plus 3

, the y-intercept would be ______.

-1

-4

3

0

The y-intercept of a graph is the point where the graph intersects the y-axis. In this case, the correct answer is 3, indicating that the graph intersects the y-axis at the point (0, 3).

Explanation

The y-intercept of a graph is the point where the graph intersects the y-axis. In this case, the correct answer is 3, indicating that the graph intersects the y-axis at the point (0, 3).

4. Identify the vertex of the given parabola.

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Explanation

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5. Which of the following quadratic functions has a graph with a y-intercept of 3?

A quadratic function in the form of y = ax^2 + bx + c will have a y-intercept when x = 0. Therefore, to find a quadratic function with a y-intercept of 3, we need to find a function where c = 3. Among the given options, the only function that satisfies this condition is y = x^2 - 2x + 3.

Explanation

A quadratic function in the form of y = ax^2 + bx + c will have a y-intercept when x = 0. Therefore, to find a quadratic function with a y-intercept of 3, we need to find a function where c = 3. Among the given options, the only function that satisfies this condition is y = x^2 - 2x + 3.

6. The vertex of this parabola shows that the __________ value of the function is _____.

Minimum; 2

Maximum; 4

Minimum; 4

Maximum; 2

The vertex of a parabola represents the highest or lowest point on the graph. In this case, the vertex is described as a maximum, indicating that the highest value of the function is represented by the vertex. The value of the function at this vertex is given as 4, meaning that the function reaches a maximum value of 4.

Explanation

The vertex of a parabola represents the highest or lowest point on the graph. In this case, the vertex is described as a maximum, indicating that the highest value of the function is represented by the vertex. The value of the function at this vertex is given as 4, meaning that the function reaches a maximum value of 4.

7. What are the solutions to the equation

open parentheses x plus 5 close parentheses open parentheses x minus 2 close parentheses equals 0

?

X = -5, 2

X = 5, -2

The given equation has two solutions, -5 and 2. This means that when x is equal to -5 or 2, the equation will hold true. Therefore, the correct answer is x = -5, 2.

Explanation

The given equation has two solutions, -5 and 2. This means that when x is equal to -5 or 2, the equation will hold true. Therefore, the correct answer is x = -5, 2.

8. What are the x-intercepts of the graph of the quadratic function

y equals x squared minus 16

?

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Explanation

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9. Find the axis of symmetry of the parabola with zeros

negative 4 space text and   end text 2

.

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Explanation

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10. Find the axis of symmetry of the parabola with zeros -1 and 6

X = 1.5

X = 2

X = 2.5

X = 3

The axis of symmetry of a parabola is the vertical line that passes through the vertex of the parabola. The vertex is the midpoint between the zeros of the parabola. In this case, the zeros are -1 and 6, so the midpoint is (6 + (-1))/2 = 5/2 = 2.5. Therefore, the axis of symmetry is x = 2.5.

Explanation

The axis of symmetry of a parabola is the vertical line that passes through the vertex of the parabola. The vertex is the midpoint between the zeros of the parabola. In this case, the zeros are -1 and 6, so the midpoint is (6 + (-1))/2 = 5/2 = 2.5. Therefore, the axis of symmetry is x = 2.5.

11. Which of the following functions is quadratic?

The quadratic function is a polynomial function of degree 2. It can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants. The function y = 2x^2 + 3x - 1 is quadratic because it is a polynomial of degree 2 and can be written in the form f(x) = 2x^2 + 3x - 1.

Explanation

The quadratic function is a polynomial function of degree 2. It can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants. The function y = 2x^2 + 3x - 1 is quadratic because it is a polynomial of degree 2 and can be written in the form f(x) = 2x^2 + 3x - 1.

12. Which of the following statements is never true?

The vertex of a parabola occurs in the first quadrant.

The graph of a quadratic function that has a maximum opens upward.

The graph of a quadratic function contains the point (0, 0).

The axis of symmetry passes through the vertex.

The statement "The graph of a quadratic function that has a maximum opens upward" is never true because a quadratic function with a maximum point opens downward. A quadratic function that opens upward has a minimum point.

Explanation

The statement "The graph of a quadratic function that has a maximum opens upward" is never true because a quadratic function with a maximum point opens downward. A quadratic function that opens upward has a minimum point.

13. What are the solutions to the equation

left parenthesis x plus 1 right parenthesis left parenthesis 2 x minus 3 right parenthesis equals 0

?

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Explanation

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14. What are the solutions to the equation

0 equals x squared plus 5 x plus 6

?

X = -3, -2

X = 5, 1

X = 5, 6

X = -6, -5

The given equation has multiple solutions for x. The solutions are x = -3 and x = -2. This means that when x is equal to -3 or -2, the equation is satisfied.

Explanation

The given equation has multiple solutions for x. The solutions are x = -3 and x = -2. This means that when x is equal to -3 or -2, the equation is satisfied.

15. Which of the following statements is always true?

The graph of a quadratic function is a straight line.

The range of a quadratic function is the set of all real numbers.

The highest power of the independent variable in a quadratic function is 2.

The vertex of a parabola occurs at the minimum value of the function.

The highest power of the independent variable in a quadratic function is always 2. This is because a quadratic function is defined by an equation of the form f(x) = ax^2 + bx + c, where the highest power of x is 2. The coefficient "a" determines the shape and direction of the parabola, and the values of "b" and "c" determine the position of the vertex and the y-intercept. Therefore, regardless of the specific values of a, b, and c, the highest power of the independent variable in a quadratic function will always be 2.

Explanation

The highest power of the independent variable in a quadratic function is always 2. This is because a quadratic function is defined by an equation of the form f(x) = ax^2 + bx + c, where the highest power of x is 2. The coefficient "a" determines the shape and direction of the parabola, and the values of "b" and "c" determine the position of the vertex and the y-intercept. Therefore, regardless of the specific values of a, b, and c, the highest power of the independent variable in a quadratic function will always be 2.

16. Which function has zeros shown in the graph?

The function that has zeros shown in the graph is the function that crosses the x-axis at those points. Zeros of a function are the values of x for which the function equals zero. In the graph, the points where the function intersects the x-axis are the zeros of the function. These points indicate the values of x where the function evaluates to zero.

Explanation

The function that has zeros shown in the graph is the function that crosses the x-axis at those points. Zeros of a function are the values of x for which the function equals zero. In the graph, the points where the function intersects the x-axis are the zeros of the function. These points indicate the values of x where the function evaluates to zero.

17. Which of the following functions has a graph with an axis of symmetry of

x equals negative 1 half

?

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Explanation

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18. What are the solutions to the equation

9 x squared minus 36 equals 0

?

X = -6, 6

X = -4, 4

X = -3, 3

X = -2, 2

The equation has multiple solutions, which are x = -2 and x = 2. These values satisfy the equation and make it true.

Explanation

The equation has multiple solutions, which are x = -2 and x = 2. These values satisfy the equation and make it true.

19. Which of the following quadratic functions has a maximum?

A quadratic function has a maximum if the coefficient of the x^2 term is negative. This is because the graph of a quadratic function with a negative coefficient of x^2 opens downwards, creating a concave shape. The highest point on the graph is the maximum. Therefore, the quadratic function with a negative coefficient of x^2 will have a maximum.

Explanation

A quadratic function has a maximum if the coefficient of the x^2 term is negative. This is because the graph of a quadratic function with a negative coefficient of x^2 opens downwards, creating a concave shape. The highest point on the graph is the maximum. Therefore, the quadratic function with a negative coefficient of x^2 will have a maximum.

20. Which of the following quadratic functions has a minimum?

A quadratic function has a minimum when the coefficient of the squared term is positive. This is because the graph of a quadratic function with a positive coefficient opens upwards, creating a "U" shape, and the lowest point of the graph is the minimum. Therefore, any quadratic function with a positive coefficient for the squared term will have a minimum.

Explanation

A quadratic function has a minimum when the coefficient of the squared term is positive. This is because the graph of a quadratic function with a positive coefficient opens upwards, creating a "U" shape, and the lowest point of the graph is the minimum. Therefore, any quadratic function with a positive coefficient for the squared term will have a minimum.

21. The vertex of a quadratic function is in the second quadrant. The related equation has no real solutions. Which of the following statements is true?

The graph opens downward.

The graph opens upward.

The y-intercept is 0.

The axis of symmetry is x = 0.

If the vertex of a quadratic function is in the second quadrant, it means that the parabola opens upward. This is because the vertex is the lowest point on the graph, and if it is in the second quadrant, it must be above the x-axis. Since the parabola opens upward, it does not intersect the x-axis, indicating that the related equation has no real solutions. Therefore, the correct statement is "The graph opens upward."

Explanation

If the vertex of a quadratic function is in the second quadrant, it means that the parabola opens upward. This is because the vertex is the lowest point on the graph, and if it is in the second quadrant, it must be above the x-axis. Since the parabola opens upward, it does not intersect the x-axis, indicating that the related equation has no real solutions. Therefore, the correct statement is "The graph opens upward."