Question

Find the equation of the axis of symmetry and the coordinates of the vertex of the graph of the following function.yequalsnegative 2 x squared plus 3 x minus 4Question content area bottomPart 1The axis of symmetry is xequalsenter your response here. ​(Simplify your​ answer.)

Answer 1

To find the axis of symmetry and the coordinates of the vertex of the quadratic function \( y = -2x^2 + 3x - 4 \), you can use the formula for the axis of symmetry, which is given by:

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

where \( a \) and \( b \) are the coefficients from the quadratic equation \( ax^2 + bx + c \). Here, \( a = -2 \), \( b = 3 \), and \( c = -4 \).

Plugging in the values of \( a \) and \( b \):

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

So the axis of symmetry is:

\[
x = \frac{3}{4}
\]

Next, to find the vertex \( (x, y) \) coordinates, substitute \( x = \frac{3}{4} \) back into the original function to find \( y \):

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

Calculating each term:

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

Now add these results together:

\[
y = -\frac{9}{8} + \frac{9}{4} - 4
\]

Convert \( \frac{9}{4} \) to eighths:

\[
\frac{9}{4} = \frac{18}{8}
\]

And \( 4 = \frac{32}{8} \). Now combining everything:

\[
y = -\frac{9}{8} + \frac{18}{8} - \frac{32}{8} = \frac{-9 + 18 - 32}{8} = \frac{-23}{8}
\]

Thus, the vertex coordinates are:

\[
\left(\frac{3}{4}, -\frac{23}{8}\right)
\]

To summarize:

- The axis of symmetry is \( x = \frac{3}{4} \).
- The coordinates of the vertex are \( \left(\frac{3}{4}, -\frac{23}{8}\right) \).