Question

for the quadratic function ( f(x)=2x^{2}-3x + 2 ), answer parts (a) through (f).
is the graph concave up or concave down?
concave up
concave down
(b) find the y - intercept and the x - intercepts, if any. what is the x - intercept? select the correct choice below and, if necessary, fill in the answer box to compete your choice.
the x - intercept(s) is/are
(type an integer or a simplified fraction. use a comma to separate answers as needed.)
there are no x - intercepts.
what is the y - intercept? select the correct choice below and, if necessary, fill in the answer box to compete your choice.
the y - intercept(s) is/are 2.
(type an integer or a simplified fraction. use a comma to separate answers as needed.)
there are no y - intercepts.
(c) use parts (a) and (b) to graph the function. choose the correct graph below.

Explanation:

Step1: Determine the 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.
In the function \(f(x)=2x^{2}-3x + 2\), \(a = 2>0\). So the graph is concave up.

Step2: Find the \(y\) - intercept

The \(y\) - intercept is found by setting \(x = 0\).
Substitute \(x = 0\) into \(f(x)=2x^{2}-3x + 2\), we get \(f(0)=2(0)^{2}-3(0)+2=2\).

Step3: Find the \(x\) - intercepts

The \(x\) - intercepts are found by setting \(y=f(x)=0\), so we solve the quadratic equation \(2x^{2}-3x + 2=0\).
The quadratic formula for \(ax^{2}+bx + c = 0\) is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
Here, \(a = 2\), \(b=-3\), \(c = 2\).
First, calculate the discriminant \(\Delta=b^{2}-4ac=(-3)^{2}-4\times2\times2=9 - 16=-7<0\).
Since \(\Delta<0\), there are no real solutions for \(x\) (no \(x\) - intercepts).

Answer:

• Concavity: Concave up
• \(x\) - intercept: There are no \(x\) - intercepts
• \(y\) - intercept: \(2\)