To find the vertex of the quadratic equation $y = 4x^2 + 3x - 8$, we can use the vertex formula. For a quadratic equation in the form $y = ax^2 + bx + c$, the x-coordinate of the vertex is:

$x = -\frac{b}{2a}.$

Here, $a = 4$, $b = 3$, and $c = -8$.

Step 1: Find the x-coordinate of the vertex

$x = -\frac{b}{2a} = -\frac{3}{2(4)} = -\frac{3}{8} = -0.375.$

Step 2: Substitute $x = -0.375$ into the quadratic equation to find the y-coordinate

$y = 4(-0.375)^2 + 3(-0.375) - 8.$

1. Compute $(-0.375)^2$: $(-0.375)^2 = 0.140625.$

2. Multiply by 4: $4(0.140625) = 0.5625.$

3. Multiply $3(-0.375)$: $3(-0.375) = -1.125.$

4. Add and subtract the terms: $y = 0.5625 - 1.125 - 8 = -8.5625.$

Thus, the vertex is approximately $(-0.375, -8.563)$.

Final Answer:

The correct choice is: $\boxed{(-0.375, -8.563)}$

Do you want more details or clarification? Let me know!

Related Questions:

1. How do you determine the axis of symmetry for a quadratic function?
2. What is the importance of the vertex in graphing quadratic functions?
3. How do you convert a quadratic equation into vertex form?
4. Can you explain how to identify if the vertex is a maximum or minimum?
5. How do you find the y-intercept of a quadratic equation?

Tip:

The vertex is the turning point of the quadratic graph. Use $x = -\frac{b}{2a}$ to quickly locate it when in standard form!