Question

graph the equation $y = -x^2 - 12x - 35$ 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 the roots (y=0)

Set $y=0$: $0 = -x^2 -12x -35$
Multiply by $-1$: $x^2 +12x +35=0$
Factor: $(x+5)(x+7)=0$
Solutions: $x=-5, x=-7$
Points: $(-7,0), (-5,0)$

Step2: Find the vertex x-coordinate

Use $x=-\frac{b}{2a}$ for $ax^2+bx+c$
Here $a=-1, b=-12$:
$x=-\frac{-12}{2(-1)} = \frac{12}{-2} = -6$

Step3: Find vertex y-coordinate

Substitute $x=-6$ into $y=-x^2-12x-35$:
$y=-(-6)^2 -12(-6)-35 = -36 +72 -35 = 1$
Vertex point: $(-6,1)$

Step4: Find 2 additional points

Choose $x=-8$:
$y=-(-8)^2 -12(-8)-35 = -64+96-35 = -3$
Point: $(-8,-3)$
Choose $x=-4$:
$y=-(-4)^2 -12(-4)-35 = -16+48-35 = -3$
Point: $(-4,-3)$

Step5: Determine axis of symmetry

The axis of symmetry is the vertical line through the vertex x-coordinate.

Answer:

Plotted Points:

$(-7,0), (-5,0), (-6,1), (-8,-3), (-4,-3)$

Equation of Axis of Symmetry:

$x=-6$