Graph the quadratic y=4x2+3x−8

Question

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

(0,−8)
(
0
,
−
8
)
left parenthesis 0 comma negative 8 right parenthesis

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

(−0.375,−8.563)
(
−
0.375
,
−
8.563
)
left parenthesis negative 0.375 comma negative 8.563 right parenthesis

(1.088,0)

Answer 1

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

The x-coordinate of the vertex is given by:

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

For our quadratic:

• \( a = 4 \)
• \( b = 3 \)
• \( c = -8 \)

Substituting the values of \( a \) and \( b \):

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

Next, we need to find the y-coordinate by substituting \( x = -\frac{3}{8} \) back into the quadratic equation:

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

Calculating each term:

1. \( 4\left(-\frac{3}{8}\right)^2 = 4 \cdot \frac{9}{64} = \frac{36}{64} = \frac{9}{16} \)
2. \( 3\left(-\frac{3}{8}\right) = -\frac{9}{8} = -\frac{18}{16} \)
3. \( -8 = -\frac{128}{16} \)

Now combine these results:

\[ y = \frac{9}{16} - \frac{18}{16} - \frac{128}{16} = \frac{9 - 18 - 128}{16} = \frac{-137}{16} \]

So the vertex of the quadratic \( y = 4x^2 + 3x - 8 \) is at:

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

This specific vertex does not appear in the provided options, but if you are looking for a close approximation from the options given or an oversight, consider that for identifying the vertex accurately, you would require checking the graphical representation or calculation to validate further or replicate with another approach as the listed choices appear mismatched with the expected calculation outcomes based on the details provided.

Thus, I'd say none of the options you provided are accurate representations of the calculated vertex coordinates through standard vertex finding methods.

If we were to consult the potential options if those denominators are considering inferences from approximations in decimal: The closest approximation for \( \left(-0.375, -8.563\right) \) if adapted could yield possible graphical error or simplifications made for easier interpretative read.

Make sure to check your calculation through verification to ensure clarity on vertex measurement and implications when drawn on graphical representation.