Question

for the quadratic function f(x)= -x² - 2x, answer parts (a) through (f).
the vertex is (-1,1). (type an ordered pair, using integers or fractions.)
what is the equation of the axis of symmetry?
the axis of symmetry is x = -1. (use integers or fractions for any numbers in the equation.)
is the graph concave up or concave down?
concave down
concave up

Explanation:

Step1: Recall quadratic - function form

For a quadratic function $f(x)=ax^{2}+bx + c$, in the given function $f(x)=-x^{2}-2x$, we have $a=-1$, $b = - 2$, $c = 0$.

Step2: Find the x - coordinate of the vertex

The formula for the x - coordinate of the vertex of a quadratic function is $x=-\frac{b}{2a}$. Substitute $a=-1$ and $b = - 2$ into the formula: $x=-\frac{-2}{2\times(-1)}=-1$.

Step3: Find the y - coordinate of the vertex

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

Step4: Determine the axis of symmetry

The equation of the axis of symmetry of a quadratic function $y = ax^{2}+bx + c$ is $x=-\frac{b}{2a}$. Since we already found that $x = - 1$ for the x - coordinate of the vertex, the equation of the axis of symmetry is $x=-1$.

Step5: 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. Here $a=-1<0$, so the graph is concave down.

Answer:

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