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.