Question

Graph the quadratic y=4x2+3x−8
𝑦
=
4
𝑥
2
+
3
𝑥
−
8
. What is the vertex of this quadratic?(1 point)
Responses

(1.088,0)
(
1.088
,
0
)
left parenthesis 1.088 comma 0 right parenthesis

(−1.838,0)
(
−
1.838
,
0
)
left parenthesis negative 1.838 comma 0 right parenthesis

(0,−8)
(
0
,
−
8

Answer 1

To find the vertex of the quadratic function \( y = 4x^2 + 3x - 8 \), we can use the vertex formula. The x-coordinate of the vertex for a quadratic in the form \( y = ax^2 + bx + c \) is given by:

\[
x = -\frac{b}{2a}
\]

For the given quadratic, \( a = 4 \) and \( b = 3 \). Plugging in these values:

\[
x = -\frac{3}{2 \cdot 4} = -\frac{3}{8}
\]

Now, to find the corresponding y-coordinate of the vertex, we substitute \( x = -\frac{3}{8} \) back into the original equation:

\[
y = 4\left(-\frac{3}{8}\right)^2 + 3\left(-\frac{3}{8}\right) - 8
\]

Calculating the terms step-by-step:

1. \(\left(-\frac{3}{8}\right)^2 = \frac{9}{64}\)
2. \(4 \cdot \frac{9}{64} = \frac{36}{64} = \frac{9}{16}\)
3. \(3 \cdot -\frac{3}{8} = -\frac{9}{8} = -\frac{72}{64}\)
4. Now combine all terms:

\[
y = \frac{9}{16} - \frac{72}{64} - 8
\]

To combine these, we need a common denominator. The common denominator for 16 and 64 is 64.

- \(\frac{9}{16} = \frac{36}{64}\)
- \(-8 = -\frac{512}{64}\)

So,

\[
y = \frac{36}{64} - \frac{72}{64} - \frac{512}{64} = \frac{36 - 72 - 512}{64} = \frac{-548}{64} = -\frac{137}{16}
\]

Now that we have the x-coordinate and y-coordinate, the vertex of the quadratic function \( y = 4x^2 + 3x - 8 \) is:

\[
\left(-\frac{3}{8}, -\frac{137}{16}\right)
\]

Since this doesn't match any of the given answer choices, let us verify if any of the choices provided could represent significant points.

Evaluating the options:

1. \((1.088, 0)\) — Not the vertex.
2. \((-1.838, 0)\) — Not the vertex.
3. \((0, -8)\) — This point is just where it crosses the y-axis, not the vertex.

Given this, it seems there may have been an error with the provided options for the vertex of this particular quadratic triplet. The vertex of the quadratic \( y = 4x^2 + 3x - 8 \) is approximately \((-0.375, -8.5625)\). If you are looking for exact points, there may be no correct option from your list.

Please check the problem prompt or the associated graph for any additional clarifications.