To convert the given quadratic equation $y = -\frac{5}{16}x^2 - \frac{5}{8}x + \frac{74}{16}$ into vertex form, we use the method of completing the square. Here's the step-by-step process:

Step 1: Factor out the coefficient of $x^2$ from the first two terms

Factor $-\frac{5}{16}$ from $-\frac{5}{16}x^2 - \frac{5}{8}x$: $y = -\frac{5}{16} \left( x^2 + 2x \right) + \frac{74}{16}$

Step 2: Complete the square inside the parentheses

The term to complete the square is found by taking half the coefficient of $x$ (which is 2), squaring it, and adding it inside the parentheses: $\left(\frac{2}{2}\right)^2 = 1$ Add and subtract 1 inside the parentheses: $y = -\frac{5}{16} \left( x^2 + 2x + 1 - 1 \right) + \frac{74}{16}$ $y = -\frac{5}{16} \left( (x+1)^2 - 1 \right) + \frac{74}{16}$

Step 3: Distribute the coefficient of the square term

Distribute $-\frac{5}{16}$ to $(x+1)^2$ and $-1$: $y = -\frac{5}{16}(x+1)^2 + \frac{5}{16} + \frac{74}{16}$

Step 4: Simplify the constant terms

Combine the constant terms: $\frac{5}{16} + \frac{74}{16} = \frac{79}{16}$ Thus: $y = -\frac{5}{16}(x+1)^2 + \frac{79}{16}$

Final Vertex Form:

The equation in vertex form is: $y = -\frac{5}{16}(x+1)^2 + \frac{79}{16}$ The vertex of the parabola is $(-1, \frac{79}{16})$.

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Would you like additional clarification or details on any step?

5 Related Questions:

1. How do you determine the vertex of a quadratic equation in vertex form?
2. What are the advantages of completing the square compared to other methods?
3. How can you graph a quadratic function using the vertex form?
4. What is the relationship between the standard form and the vertex form of a quadratic equation?
5. Can the process of completing the square be used to solve quadratic equations?

Tip:

When completing the square, always double-check your calculations for added or subtracted terms to ensure accuracy!