Question

graph the equation $y = -x^2 - 14x - 48$ on the accompanying set of axes. you must plot 5 points including the roots and the vertex. using the graph, determine the equation of the axis of symmetry.
click to plot points. click points to delete them.

Explanation:

Step1: Find roots (y=0)

Set $-x^2 -14x -48 = 0$, multiply by $-1$: $x^2 +14x +48=0$. Factor: $(x+6)(x+8)=0$. Roots: $x=-6, x=-8$. Points: $(-8,0), (-6,0)$

Step2: Find vertex x-coordinate

Use $x=-\frac{b}{2a}$. For $y=-x^2-14x-48$, $a=-1, b=-14$.
$x=-\frac{-14}{2(-1)} = -7$

Step3: Find vertex y-coordinate

Substitute $x=-7$ into equation:
$y=-(-7)^2 -14(-7) -48 = -49 +98 -48 = 1$. Vertex: $(-7,1)$

Step4: Find 2 extra points

Choose $x=-9$: $y=-(-9)^2-14(-9)-48=-81+126-48=-3$. Point: $(-9,-3)$
Choose $x=-5$: $y=-(-5)^2-14(-5)-48=-25+70-48=-3$. Point: $(-5,-3)$

Step5: Find axis of symmetry

Axis of symmetry is vertical line through vertex x-coordinate: $x=-7$

Answer:

Points to plot: $(-9,-3), (-8,0), (-7,1), (-6,0), (-5,-3)$
Equation of the axis of symmetry: $x=-7$