1. Let f(x)=x^2+4x Which equation among the following is the correct result after setting f(x) equal to 0 and completing the square?.
A. (x+2)^2−4=0
B. (x−2)^2+4=0
C. (x−2)^2−4=0
D. (x+2)^2+4=0
2. The equation (x−1)^2−9=0 is obtained after setting a function f(x) equals 0 and completing the square. What is the vertex and axis of symmetry for f(x)?
A. The vertex is located at (1,9) and the axis of symmetry is at x=1
B. The vertex is located at (1,−9) and the axis of symmetry is at x=1 .
C. The vertex is located at (1,−9) and the axis of symmetry is at x=−1 .
D. The vertex is located at (−1,−9) and the axis of symmetry is at x=−1
3. Write the quadratic function f(x)=2x^2+4x−3
A. f(x)=(x+1)^2−4
B. f(x)=2(x+1)^2−5
C. f(x)=2(x−1)^2−5
D. f(x)=2(x+1)^2−4
4. Which function among the following has an axis of symmetry at x=3 and vertex at (3, 4)
A. f(x)=2(x-3) ^2+4
B. f(x)=1/2(x+3) ^2-4
C. f(x)=1/2(x-3) ^2-4
D. f(x)=-2(x-3) ^2+4
5. Which of the following statements is true about the quadratic function f(x)=x^2+8x−9
A. The vertex is located at (−4,−25) .
B. The equation (x+4)^2−7=0 is the result of setting f(x)=0 and completing the square.
C. The axis of symmetry is at x=4
D. The parabola created by f(x) opens downard.
1. To find the equation after setting f(x) equal to 0 and completing the square, we start with the equation f(x) = x^2 + 4x.
Step 1: Set the equation equal to 0: x^2 + 4x = 0.
Step 2: Complete the square by adding (4/2)^2 = 4 to both sides of the equation: x^2 + 4x + 4 = 4.
Step 3: Factor the left side of the equation: (x + 2)^2 = 4.
So, the correct equation after setting f(x) equal to 0 and completing the square is (x + 2)^2 − 4 = 0. Therefore, the answer is A.
2. The given equation is (x - 1)^2 - 9 = 0. To find the vertex and axis of symmetry for the function f(x), we compare it with the standard form of a quadratic equation, which is f(x) = a(x - h)^2 + k, where (h, k) represents the vertex.
Comparing the given equation with the standard form, we see that h = 1 and k = 9. Therefore, the vertex is located at (1, 9).
To find the axis of symmetry, we take the x-coordinate of the vertex. So, the axis of symmetry is at x = 1.
Therefore, the correct answer is A. The vertex is located at (1, 9) and the axis of symmetry is at x = 1.
3. Given the quadratic function f(x) = 2x^2 + 4x - 3, we want to rewrite it in the form f(x) = a(x - h)^2 + k.
To do that, we follow these steps:
Step 1: Factor out the coefficient of x^2: f(x) = 2(x^2 + 2x) - 3.
Step 2: Complete the square inside the parentheses: f(x) = 2(x^2 + 2x + 1 - 1) - 3.
Step 3: Factor the square trinomial and simplify: f(x) = 2((x + 1)^2 - 1) - 3.
Step 4: Distribute and simplify: f(x) = 2(x + 1)^2 - 2 - 3.
Step 5: Combine like terms: f(x) = 2(x + 1)^2 - 5.
So, the correct answer is B. f(x) = 2(x + 1)^2 - 5.
4. To find the function with an axis of symmetry at x = 3 and a vertex at (3, 4), we use the standard form f(x) = a(x - h)^2 + k, where (h, k) represents the vertex.
Comparing the given information with the standard form, we see that h = 3 and k = 4.
Since the axis of symmetry is at x = 3, the equation has to be of the form f(x) = a(x - 3)^2 + 4.
To find the value of a, we substitute the coordinates of the vertex, (3, 4), into the equation.
4 = a(3 - 3)^2 + 4.
4 = a(0) + 4.
4 = 4a.
a = 1.
Therefore, the correct equation is f(x) = (x - 3)^2 + 4, which is equivalent to A.
5. To determine the properties of the quadratic function f(x) = x^2 + 8x - 9, let's analyze each statement.
A. The vertex is located at (-4, -25).
We can check the vertex using the formula x = -b/2a and substituting the values from the quadratic function. In this case, x = -8/(2*1) = -4.
Substituting x = -4 into f(x) gives us:
f(-4) = (-4)^2 + 8(-4) - 9 = 16 - 32 - 9 = -25.
So, the vertex is located at (-4, -25).
B. The equation (x + 4)^2 - 7 = 0 is the result of setting f(x) = 0 and completing the square.
To check if this statement is true, we set f(x) = 0 and complete the square:
x^2 + 8x - 9 = 0.
To complete the square, add (8/2)^2 = 16 to both sides of the equation:
x^2 + 8x + 16 - 9 = 16.
Rearranging the equation:
(x + 4)^2 - 9 = 16.
(x + 4)^2 - 7 = 0.
So, this statement is true.
C. The axis of symmetry is at x = 4.
The axis of symmetry for a quadratic function is given by the formula x = -b/2a. From the given quadratic function, a = 1 and b = 8.
-x = -8/(2*1) = -4.
Therefore, the axis of symmetry is at x = -4, not x = 4. So, this statement is false.
D. The parabola created by f(x) opens downward.
To determine if the parabola opens upward or downward, we can examine the coefficient of x^2. In this case, the coefficient is positive (1), which means the parabola opens upward.
Therefore, the correct statement is: The parabola created by f(x) opens upward. (D is false).
So, the correct answer is B. The equation (x + 4)^2 - 7 = 0 is the result of setting f(x) = 0 and completing the square.
1. The correct equation after setting f(x) equal to 0 and completing the square is A. (x+2)^2−4=0.
2. The vertex is located at (1,9) and the axis of symmetry is at x=1. The correct answer is A.
3. The quadratic function f(x)=2x^2+4x−3 can be rewritten as f(x)=(x+1)^2−4. The correct answer is A.
4. The function with an axis of symmetry at x=3 and a vertex at (3,4) is f(x)=1/2(x-3)^2-4. The correct answer is C.
5. The statement that is true about the quadratic function f(x)=x^2+8x−9 is C. The axis of symmetry is at x=4.
1. A. (x+2)^2−4=0
2. A. The vertex is located at (1,9) and the axis of symmetry is at x=1
3. D. f(x)=2(x+1)^2−4
4. C. f(x)=1/2(x-3) ^2-4
5. B. The equation (x+4)^2−7=0 is the result of setting f(x)=0 and completing the square.