Question

for the quadratic function f(x)= -x² - 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 - formula for quadratic function

For a quadratic function $y = ax^{2}+bx + c$, the $x$ - coordinate of the vertex is $x=-\frac{b}{2a}$. Given $f(x)=-x^{2}-2x$, where $a=-1$ and $b = - 2$. Then $x=-\frac{-2}{2\times(-1)}=-1$.

Step2: Find the $y$ - coordinate of the vertex

Substitute $x = - 1$ into the function $f(x)=-x^{2}-2x$. So $f(-1)=-(-1)^{2}-2\times(-1)=-1 + 2=1$.

Step3: Determine the axis of symmetry

The axis of symmetry of a quadratic function $y = ax^{2}+bx + c$ is the vertical line $x=-\frac{b}{2a}$. So the axis of symmetry is $x=-1$.

Step4: Determine concavity

Since $a=-1<0$, the graph of the quadratic function is concave down.

Answer:

$(-1,1)$