1.
Which of the following functions is quadratic?
Correct Answer
B.
Explanation
A quadratic function is a function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a â‰ 0. The given options are: a) f(x) = 3x^2 - 5x + 2, b) f(x) = 4x^3 - 2x + 1, c) f(x) = 2x + 3, and d) f(x) = x^2 + 4x + 2. Option d) f(x) = x^2 + 4x + 2 is quadratic because it is in the form of a quadratic function with a â‰ 0. Option a) is also quadratic, but option d) is the only correct answer.
2.
The vertex of this parabola shows that the __________ value of the function is _____.
Correct Answer
B. Maximum; 4
Explanation
The vertex of a parabola represents the highest or lowest point on the graph. In this case, the vertex is at its highest point, indicating a maximum value. The y-coordinate of the vertex is 4, which means that the maximum value of the function is 4.
3.
Which of the following statements is SOMETIMES true?
Correct Answer
D. The vertex of a parabola occurs at the minimum value of the function.
Explanation
The statement "The vertex of a parabola occurs at the minimum value of the function" is sometimes true because it depends on the shape of the parabola. If the parabola opens upward, then the vertex represents the minimum value of the function. However, if the parabola opens downward, then the vertex represents the maximum value of the function. Therefore, the statement is only true for parabolas that open upward.
4.
Which of the following quadratic functions has a maximum?
Correct Answer
A.
Explanation
A quadratic function has a maximum when the coefficient of the squared term is negative. This is because the graph of a quadratic function with a negative coefficient for the squared term opens downwards, creating a "U" shape. The vertex of this "U" shape represents the maximum point of the function. Therefore, the quadratic function with a negative coefficient for the squared term will have a maximum.
5.
Which of the following quadratic functions has a minimum?
Correct Answer
D.
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 for the squared term opens upwards, creating a "U" shape. The vertex of this "U" shape represents the minimum point of the function. Therefore, the quadratic function with a positive coefficient for the squared term will have a minimum.
6.
Identify the vertex of the given parabola.
Correct Answer
A.
7.
The vertex of this parabola shows that the __________ value of the function is _____.
Correct Answer
C. Minimum; -9
Explanation
The vertex of a parabola represents the highest or lowest point on the graph, depending on the direction of the curve. In this case, the vertex represents the minimum value of the function. The y-coordinate of the vertex is -9, indicating that the function reaches its lowest point at that value.
8.
Which of the following functions is quadratic?
Correct Answer
D.
Explanation
A quadratic function is a function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a is not equal to zero. The given options are f(x) = 2x^2 + 3x + 1, f(x) = 4x + 5, f(x) = x^3 + 2x^2 + x, and f(x) = 6x^2 - 2x + 3. Among these options, only f(x) = 2x^2 + 3x + 1 is quadratic because it follows the form of a quadratic function with a non-zero coefficient for the x^2 term.
9.
Given the standard form of a quadratic function, , which of the following indicates a parabola that opens upward?
Correct Answer
B.
Explanation
A quadratic function in standard form is written as f(x) = ax^2 + bx + c, where a, b, and c are constants. In this form, the coefficient of the x^2 term (a) determines the direction of the parabola. If a is positive, the parabola opens upward. Therefore, the correct answer would be the option that has a positive coefficient for the x^2 term.
10.
Given the standard form of a quadratic function, , which of the following indicates a parabola that opens downward?
Correct Answer
C.
Explanation
The standard form of a quadratic function is y = ax^2 + bx + c. In this form, the coefficient 'a' determines the direction of the parabola. If 'a' is negative, the parabola opens downward. Therefore, the correct answer would be the option that has a negative coefficient for 'a'.
11.
Tell whether the following statement is SOMETIMES, ALWAYS, or NEVER true.The graph of a quadratic function is a straight line.
Correct Answer
C. Never true
Explanation
The graph of a quadratic function is never a straight line because a quadratic function is a polynomial function of degree 2, which means it has a squared term. The graph of a quadratic function is always a curve, either concave up or concave down, and never a straight line.
12.
Tell whether the following statement is SOMETIMES, ALWAYS, or NEVER true.The range of a quadratic function is the set of all real numbers.
Correct Answer
C. Never true
Explanation
A quadratic function is a function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a â‰ 0. The range of a quadratic function depends on the value of the coefficient a. If a > 0, the range is the set of all real numbers greater than or equal to the vertex of the parabola. If a < 0, the range is the set of all real numbers less than or equal to the vertex. Therefore, the range of a quadratic function is never the set of all real numbers.
13.
Tell whether the following statement is SOMETIMES, ALWAYS, or NEVER true.The highest power of the independent variable in a quadratic function is 2.
Correct Answer
B. Always true
Explanation
In a quadratic function, the highest power of the independent variable is always 2. This is because a quadratic function is defined as a function of the form f(x) = ax^2 + bx + c, where the highest power of x is 2. Therefore, it is always true that the highest power of the independent variable in a quadratic function is 2.
14.
Tell whether the following statement is SOMETIMES, ALWAYS, or NEVER true.The graph of a quadratic function that has a minimum opens upward.
Correct Answer
B. Always true
Explanation
The statement is always true because the graph of a quadratic function that has a minimum point always opens upward. This is because the coefficient of the quadratic term is positive, which causes the graph to form a U-shape with the vertex at the minimum point. Regardless of the specific values of the coefficients, the graph will always exhibit this upward-opening behavior.
15.
Which of the following is true of a parabola?
Correct Answer
D. All of the above
Explanation
A parabola is the shape of the graph of a quadratic function. It can have a low point (minimum) or a high point (maximum), depending on the direction it opens. Additionally, a parabola can be divided into two identical, though mirror-image, halves. Therefore, all of the given statements are true of a parabola.
16.
Which of the following functions has a graph with an axis of symmetry of ?
Correct Answer
D.
17.
What is the definition of the x-intercept?
Correct Answer
B. The value of x when y=0
Explanation
The x-intercept is the point on the graph where the function crosses the x-axis. At this point, the value of y is equal to zero. Therefore, the correct definition of the x-intercept is "the value of x when y=0".
18.
Which of the following is the equivalent of a zero of a function?
Correct Answer
C. The x-intercept
Explanation
The x-intercept of a function is the point where the graph of the function intersects the x-axis. At this point, the value of the function is equal to zero. Therefore, the x-intercept is equivalent to a zero of a function.
19.
What is the maximum number of zeros a quadratic function can have?
Correct Answer
B. 2
Explanation
A quadratic function is defined by an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants. The maximum number of zeros a quadratic function can have is 2. This is because a quadratic function represents a parabola, which can intersect the x-axis at most twice. The zeros of the function correspond to the x-intercepts of the parabola. Therefore, the correct answer is 2.
20.
What does it mean for a quadratic function to have no zeros?
Correct Answer
A. The grapH does not touch or pass through the x-axis
Explanation
When a quadratic function has no zeros, it means that the graph of the function does not intersect or touch the x-axis. In other words, there are no values of x for which the function equals zero. This can be visualized as a parabola that does not cross the x-axis and remains either entirely above or entirely below it. The other options provided in the question are not accurate explanations for a quadratic function having no zeros.
21.
Find the zeros of from its graph below.
Correct Answer
A. 2 and 6
Explanation
The graph of the given equation intersects the x-axis at 2 and 6. These are the points where the value of the equation is equal to zero, indicating that these are the zeros of the equation.
22.
What is the definition of the axis of symmetry?
Correct Answer
D. A vertical line that divides a parabola into two symmetrical halves
Explanation
The axis of symmetry is a vertical line that divides a parabola into two symmetrical halves. It is not the value of x when y=0, the highest power of the independent variable, or the maximum or minimum of the function.
23.
Which of the following is the best method for finding the axis of symmetry given the zeros of the function?
Correct Answer
B. Add the two zeros and divide by 2
Explanation
To find the axis of symmetry, we need to determine the x-coordinate of the vertex of the parabola. Since the zeros of the function are symmetrically located on either side of the vertex, the axis of symmetry will pass through the midpoint of these zeros. Adding the two zeros and dividing by 2 gives us the x-coordinate of the axis of symmetry. Therefore, adding the two zeros and dividing by 2 is the best method for finding the axis of symmetry.
24.
Find the axis of symmetry of the parabola with zeros 2 and 12.
Correct Answer
B. X = 7
Explanation
The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola. In this case, since the zeros of the parabola are 2 and 12, the vertex lies exactly in the middle of these two zeros. Therefore, the x-coordinate of the vertex is the average of the two zeros, which is (2 + 12) / 2 = 7. Hence, the equation of the axis of symmetry is x = 7.
25.
Find the axis of symmetry of the parabola with zeros .
Correct Answer
A.
26.
Find the axis of symmetry of the parabola with zeros -1 and 6
Correct Answer
C. X = 2.5
Explanation
The axis of symmetry of a parabola is a 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. The midpoint between -1 and 6 is (6 - 1) / 2 = 2.5. Therefore, the axis of symmetry of the parabola is x = 2.5.
27.
What is the formula for finding the axis of symmetry?
Correct Answer
A.
Explanation
The formula for finding the axis of symmetry of a quadratic function is x = -b/2a, where a, b, and c are coefficients of the quadratic equation in the form ax^2 + bx + c. This formula is derived from completing the square method and represents the x-coordinate of the vertex of the parabola.
28.
Find the axis of symmetry of the graph of the function .
Correct Answer
C. X = â€“2
Explanation
The correct answer is x = â€“2 because the axis of symmetry is a vertical line that divides the graph into two equal halves. In this case, the line x = â€“2 will divide the graph of the function into two equal halves, with the same values of y on either side of the line.
29.
Find the axis of symmetry of the graph of the function .
Correct Answer
D. X = 4
Explanation
The axis of symmetry of a graph is a vertical line that divides the graph into two symmetric parts. In this case, the given answer x = 4 represents the axis of symmetry. This means that the graph of the function will be symmetric with respect to the line x = 4.
30.
Find the vertex of the graph of the function .
Correct Answer
B. (â€“3, â€“25)
Explanation
The vertex of a graph represents the highest or lowest point on the graph. In this case, the vertex is represented by the point (-3, -25), which means that the function reaches its lowest point at x = -3 and y = -25. This point is the lowest point on the graph and is also known as the minimum point.
31.
Find the vertex of the graph of the function .
Correct Answer
A. (â€“1, 9)
Explanation
The vertex of a graph represents the highest or lowest point on the graph. In this case, the vertex is given as (-1, 9), which means that the graph reaches its highest point at x = -1 and y = 9.
32.
Choose the function that best represents the graph.
Correct Answer
B.
33.
Which of the following is true of the graph of the function ?
Correct Answer
A. The axis of symmetry is x = 1.
Explanation
The axis of symmetry of a parabola is a vertical line that divides the parabola into two equal halves. In this case, the axis of symmetry is x = 1, which means that the parabola is symmetric with respect to the line x = 1. This means that if you reflect any point on one side of the parabola across the axis of symmetry, you will get a corresponding point on the other side of the parabola.
34.
Which of the following is true of the graph of the function ?
Correct Answer
B. The y-intercept is +5.
Explanation
The y-intercept of a function is the point where the graph intersects the y-axis. In this case, the y-intercept is given as +5, which means that the graph intersects the y-axis at the point (0, 5). This information tells us that when x = 0, the value of the function is 5. Therefore, the statement "The y-intercept is +5" is true.
35.
For a quadratic function written in the form , when x = 0, y = c. So the y-intercept of a quadratic function is ________.
Correct Answer
C. C
Explanation
When a quadratic function is written in the form y = ax^2 + bx + c, the value of c represents the y-intercept. This is because the y-intercept occurs when x = 0, and substituting x = 0 into the equation gives y = c. Therefore, the correct answer is c.
36.
Because a parabola is symmetrical, each point is the same number of units away from the axis of symmetry as its reflected point.
Correct Answer
A. True
Explanation
A parabola is a symmetrical curve that has an axis of symmetry. This means that if you take any point on the parabola and draw a line perpendicular to the axis of symmetry, the distance from that point to the axis of symmetry will be the same as the distance from its reflected point on the other side of the axis. Therefore, the statement "Because a parabola is symmetrical, each point is the same number of units away from the axis of symmetry as its reflected point" is true.
37.
A graph of a quadratic function has points both above and below the vertex.
Correct Answer
B. False
Explanation
The statement is false because a graph of a quadratic function has points either above or below the vertex, but not both. The vertex is the highest or lowest point on the graph, depending on whether the quadratic function opens upwards or downwards. Therefore, the graph can only have points either above or below the vertex, but not both.
38.
Which of the following gives the best argument for factoring a quadratic function before attempting to graph it?
Correct Answer
A. The factors provide the x-intercepts of the grapH.
Explanation
Factoring a quadratic function allows us to find the x-intercepts or roots of the graph. By factoring the quadratic function, we can determine the values of x for which the function equals zero. These x-intercepts are important because they indicate the points where the graph crosses the x-axis. Therefore, knowing the factors of a quadratic function is crucial for accurately graphing it.
39.
What are the x-intercepts of the graph of the function ?
Correct Answer
A. 2 and 5
Explanation
The x-intercepts of a graph are the points where the graph intersects the x-axis. In this case, the x-intercepts are given as 2 and 5. This means that the graph of the function crosses the x-axis at x = 2 and x = 5.
40.
What are the x-intercepts of the graph of the function ?
Correct Answer
D. â€“3 and 4
Explanation
The x-intercepts of a graph are the points where the graph intersects the x-axis. In this case, the x-intercepts are -3 and 4. This means that when the function is graphed, the points (-3, 0) and (4, 0) will be on the graph, indicating that the function crosses the x-axis at these points.
41.
Solve the equation.
Correct Answer
B. â€“8 and 4
Explanation
The given answer of â€“8 and 4 is correct because when we solve the equation, we find that the values of x that satisfy the equation are â€“8 and 4.
42.
Solve the equation.
Correct Answer
C. â€“7 and 13
43.
Solve the equation.
Correct Answer
D. The correct solution is not here
44.
Complete the square for the given expression. Write the resulting expression as a binomial squared.
Correct Answer
C.
45.
Complete the square for the given expression. Write the resulting expression as a binomial squared.
Correct Answer
D. The correct answer is not here
46.
Complete the square for the given expression. Write the resulting expression as a binomial squared.
Correct Answer
A.
47.
Solve the equation by completing the square.
Correct Answer
A.
48.
Solve the equation by completing the square.
Correct Answer
B.
49.
Solve the equation by completing the square.
Correct Answer
D. Correct solution is not here
50.
Solve the equation by completing the square.
Correct Answer
C.