To find the vertex, axis of symmetry, maximum or minimum value, and range of a quadratic equation, we can use a process called completing the square. Here's how you can solve it step by step:
Step 1: Start with the equation y = -5(x + 2)^2 - 8.
Step 2: Identify the values of a, b, and c in the general form of a quadratic equation, y = ax^2 + bx + c. In this case, a = -5, b = 0, and c = -8.
Step 3: The vertex form of a quadratic equation is y = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex. We need to convert our equation to this form.
Step 4: Expand the equation y = -5(x + 2)^2 - 8.
y = -5(x^2 + 4x + 4) - 8
y = -5x^2 - 20x - 20 - 8
y = -5x^2 - 20x - 28
Step 5: Now, we want to complete the square. To do this, focus on the terms involving x (i.e., -20x). Divide the coefficient of x by two and square it. In this case, (-20/2)^2 = 100.
Step 6: Add and subtract the value obtained from step 5 inside the parentheses. Since we divided the coefficient of x by two, we have to multiply the constant term by the same value. So, we add +100 to the equation.
y = -5x^2 - 20x + 100 - 100 - 28
Step 7: Group the squares together and simplify.
y = -5(x^2 + 4x + 100) - 128
Step 8: The equation inside the parentheses, x^2 + 4x + 100, can be rewritten as (x + 2)^2.
Step 9: Replace the equation inside the parentheses with (x + 2)^2.
y = -5(x + 2)^2 - 128
Step 10: Finally, we can identify the vertex, axis of symmetry, maximum or minimum value, and range based on the equation in vertex form.
- The vertex is given by the coordinates (-h, k) from the equation, which in this case is (-2, -128).
- The axis of symmetry is the vertical line x = -h, so the axis of symmetry is x = -2.
- The coefficient "a" in the equation (-5 in our example) determines whether the parabola opens upwards or downwards. Since a is negative, the parabola opens downwards, so the vertex represents the maximum value.
- The maximum or minimum value is given by the value of "k" in the vertex form, which is -128 in this case.
- The domain is all real numbers, as there are no restrictions on x.
- The range is all real numbers less than or equal to the y-coordinate of the vertex, which means the range is all real numbers less than or equal to -128.
So, the vertex is (-2, -128), the axis of symmetry is x = -2, the maximum value is -128, and the range is all real numbers less than or equal to -128.