Determine the standard form of a quadratic function whose graph has a y intercept of 8 and a vertex at (-2,-4). The answer is f(x)= 3x^2 + 12x +8 but I can't figure out how to get the answer.
Solve x^2 + x=6, by factoring. The answer is x=-3,2 but I can't figure out how to get the answer.
Solve 2x^2 - 5= 3x, using the quadratic formula. The answer is x=-1, 2.5 but I can't figure out how to get the answer.
Than you Reiny for your help. And yes, I am in this course. I figured out the answer, I wanted to make sure that I was doing the correct steps.
To determine the standard form of a quadratic function given its y-intercept and vertex, follow these steps:
1. Start with the general form of a quadratic function: f(x) = ax^2 + bx + c.
2. Use the fact that the vertex of the parabola is given by (-2, -4) to find the value of "a".
- Since the x-coordinate of the vertex is -2, it gives us the value of "h" in the equation x = h, which is -2.
- Set x = -2 in the equation and solve for "a": -4 = a(-2)^2 + b(-2) + c.
- Substitute the known y-coordinate of the vertex (-4) and simplify: -4 = 4a - 2b + c.
3. Use the fact that the y-intercept is 8 to find the value of "c".
- Since the y-intercept occurs when x = 0, we substitute x = 0 in the equation: 8 = a(0)^2 + b(0) + c.
- Simplify: 8 = c.
- Therefore, c = 8.
4. Substitute the value of "c" into the equation from step 2 and simplify: -4 = 4a - 2b + 8.
- Rearrange the equation: -12 = 4a - 2b.
- Divide through by 2: -6 = 2a - b.
- Rearrange the equation: b = 2a + 6.
5. Substitute the value of "c" into the equation from step 3 and simplify: 8 = c.
- Substitute c = 8.
- Rearrange the equation: 0 = 2a + 6.
- Subtract 6 from both sides: -6 = 2a.
- Divide through by 2: -3 = a.
6. Substitute the values of "a" and "c" back into the general form of a quadratic function:
- f(x) = ax^2 + bx + c becomes f(x) = (-3)x^2 + (2a + 6)x + 8.
- Simplify: f(x) = -3x^2 + (2(-3) + 6)x + 8.
- Simplify further: f(x) = -3x^2 + (0)x + 8.
- The standard form is f(x) = -3x^2 + 8.
To solve the equation x^2 + x = 6 by factoring, follow these steps:
1. Rewrite the equation in standard quadratic form: x^2 + x - 6 = 0.
2. Notice that the equation can be factored into two binomials by finding two numbers with a product of -6 and a sum of 1.
- The numbers that satisfy these requirements are -3 and 2.
3. Rewrite the equation factored: (x - 3)(x + 2) = 0.
4. Use the zero-product property: Set each factor equal to zero and solve for "x".
- Set x - 3 = 0 and solve for "x": x = 3.
- Set x + 2 = 0 and solve for "x": x = -2.
5. Therefore, the solutions to the equation x^2 + x = 6 are x = -2 and x = 3.
To solve the equation 2x^2 - 5 = 3x using the quadratic formula, follow these steps:
1. Start with the equation in standard quadratic form: 2x^2 - 3x - 5 = 0.
2. Identify the values of "a", "b", and "c" in the quadratic formula: a = 2, b = -3, c = -5.
3. Substitute these values into the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
4. Calculate the discriminant, which is b^2 - 4ac: (-3)^2 - 4(2)(-5) = 9 + 40 = 49.
5. Since the discriminant is positive, there are two distinct solutions.
6. Substitute the values into the quadratic formula: x = (-(-3) ± √(49)) / (2(2)).
- Simplify: x = (3 ± 7) / 4.
7. Solve for "x":
- When x = (3 + 7) / 4, x = 10 / 4 = 2.5.
- When x = (3 - 7) / 4, x = -4 / 4 = -1.
8. Therefore, the solutions to the equation 2x^2 - 5 = 3x are x = -1 and x = 2.5.
To determine the standard form of a quadratic function given the y-intercept and vertex, we can use the vertex form of the quadratic equation and then convert it to standard form.
1. Start with the vertex form of a quadratic function:
f(x) = a(x - h)^2 + k
where (h, k) represents the coordinates of the vertex.
2. Substitute the known values:
f(x) = a(x + 2)^2 - 4
3. Use the y-intercept to solve for 'a':
When x = 0, y = 8:
8 = a(0 + 2)^2 - 4
8 = 4a - 4
12 = 4a
a = 3
4. Substitute the value of 'a' back into the equation:
f(x) = 3(x + 2)^2 - 4
Expand and simplify:
f(x) = 3(x^2 + 4x + 4) - 4
f(x) = 3x^2 + 12x + 12 - 4
f(x) = 3x^2 + 12x + 8
Therefore, the standard form of the quadratic function is f(x) = 3x^2 + 12x + 8.
To solve the quadratic equation x^2 + x = 6 by factoring, we can rearrange the equation to have zero on one side:
1. Start with the equation:
x^2 + x - 6 = 0
2. Factor the equation:
(x + 3)(x - 2) = 0
3. Set each factor equal to zero and solve for x:
x + 3 = 0 --> x = -3
x - 2 = 0 --> x = 2
Hence, the solutions to the quadratic equation are x = -3 and x = 2.
To solve the quadratic equation 2x^2 - 5 = 3x using the quadratic formula, we can follow these steps:
1. Start with the equation:
2x^2 - 5 = 3x
2. Move all terms to one side:
2x^2 - 3x - 5 = 0
3. Identify the coefficients:
a = 2, b = -3, c = -5
4. Apply the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting in the values:
x = (-(-3) ± √((-3)^2 - 4*2*(-5))) / (2*2)
x = (3 ± √(9 + 40)) / 4
x = (3 ± √49) / 4
x = (3 ± 7) / 4
5. Simplify:
x = (3 + 7) / 4 --> x = 10 / 4 --> x = 2.5
x = (3 - 7) / 4 --> x = -4 / 4 --> x = -1
Therefore, the solutions to the quadratic equation using the quadratic formula are x = -1 and x = 2.5.
This question and the other one I just answered for you are fundamental questions that you just have to know how to do in this topic.
To simply say, "I can't figure out how to get the answer" tells me you are not even in the course.
Anyway....
since you know the vertex, the equation must be
y = a(x+2)^2 - 4
since (0,8) lies on it ....
8 = a(2^2) - 4
12 = 4a
a = 3
f(x) = 3(x+2)^2 - 4
I will leave it up to you to expand it and get the desired answer
x^2 + x=6
x^2 + x - 6 = 0
(x+3)(x-2) = 0
and ......
2x^2 - 5= 3x
2x^2 - 3x - 5 = 0
a = 2, b = -3, c = -5
now just grind it out