Question

for the quadratic function f(x)=x^{2}+2x, 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 . (type an ordered pair, using integers or fractions.)

Explanation:

Step1: Recall vertex - form of quadratic function

The general form of a quadratic function is $f(x)=ax^{2}+bx + c$, and its vertex - form is $f(x)=a(x - h)^{2}+k$, where the vertex is $(h,k)$ and the axis of symmetry is $x = h$. For the function $f(x)=x^{2}+2x$, we have $a = 1$, $b = 2$, $c = 0$. We can complete the square.

Step2: Complete the square

\[

$$\begin{align*} f(x)&=x^{2}+2x\\ &=x^{2}+2x + 1-1\\ &=(x + 1)^{2}-1 \end{align*}$$

\]

Step3: Identify the vertex

Comparing $f(x)=(x + 1)^{2}-1$ with $f(x)=a(x - h)^{2}+k$, we get $h=-1$ and $k = - 1$. So the vertex is $(-1,-1)$.

Step4: Determine the axis of symmetry

The axis of symmetry is given by the formula $x=h$. Since $h=-1$, the axis of symmetry is $x=-1$.

Step5: Determine concavity

Since $a = 1>0$, the graph of the quadratic function $y = f(x)$ is concave up.

Answer:

The vertex is $(-1,-1)$; the axis of symmetry is $x=-1$; the graph is concave up.