Question

for the quadratic function ( f(x)=2 x^{2}-3 x + 2 ), answer parts (a) through (f).
(a) find the vertex and the axis of symmetry of the quadratic function, and determine whether the graph is concave up or concave down.
the vertex is ( left(\frac{3}{4}, \frac{7}{8}
ight) )
(simplify your answer. type an ordered pair, using integers or fractions.)
what is the equation of the axis of symmetry?
the axis of symmetry is ( x=\frac{3}{4} ).
(type an equation. use integers or fractions for any numbers in the equation.)
is the graph concave up or concave down?
concave up
concave down

Explanation:

Step1: Recall the formula for the axis of symmetry

For a quadratic function \(y = ax^{2}+bx + c\), the formula for the axis of symmetry is \(x=-\frac{b}{2a}\).
In the function \(f(x)=2x^{2}-3x + 2\), \(a = 2\) and \(b=-3\).
Substitute \(a\) and \(b\) into the formula: \(x =-\frac{-3}{2\times2}=\frac{3}{4}\).

Step2: Recall the vertex formula

The \(x\) - coordinate of the vertex is the axis of symmetry \(x=\frac{3}{4}\).
To find the \(y\) - coordinate of the vertex, substitute \(x = \frac{3}{4}\) into the function \(f(x)\):
\(f(\frac{3}{4})=2\times(\frac{3}{4})^{2}-3\times\frac{3}{4}+2\)
\(=2\times\frac{9}{16}-\frac{9}{4}+2\)
\(=\frac{9}{8}-\frac{9}{4}+2\)
\(=\frac{9 - 18+16}{8}=\frac{7}{8}\).

Step3: Determine concavity

For a quadratic function \(y = ax^{2}+bx + c\), if \(a>0\), the graph is concave up; if \(a < 0\), the graph is concave down.
Since \(a = 2>0\) in \(f(x)=2x^{2}-3x + 2\), the graph is concave up.

Answer:

The vertex is \((\frac{3}{4},\frac{7}{8})\), the axis of symmetry is \(x=\frac{3}{4}\), and the graph is concave up.