Question

Your graph and the method of graphing shown and reasoningdiscussed.Labeled key features: 2-intercepts/solutions, vertex, axis ofsymmetry, y-intercept Domain and Range should be listed. Accurately labeled and appropriate scale used. Please Interpret the vertex (state max/min and apply to context). Interpret the solutions (2-intercepts) in context.

Answer 1

Quadratic function in vertex form:

$y=a(x-h)^2+k$

Vertex: (h,k)

Given function:

$y=-4(x-5)^2+125$Vertex: (5,125)

Axis of symmetry (x-value in the vertex): x=5

y-intercept: find y when x is 0:

$\begin{gathered} y=-4(0-5)^2+125 \\ y=-4(-5)^2+125 \\ y=-4(25)+125 \\ y=-100+125 \\ y=25 \end{gathered}$

y-intercept: (0,25)

x-intercepts (solutions), equal the function to zero and solve x.

$\begin{gathered} -4(x-5)^2+125=0 \\ -4(x^2-10x+25)+125=0 \\ -4x^2+40x-100+125=0 \\ -4x^2+40x+25=0 \\ \\ ax^2+bx+c=0 \\ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \\ \\ x=\frac{-40\pm\sqrt{40^2-4(-4)(25)}}{2(-4)} \\ \\ x=\frac{-40\pm\sqrt{1600+400}}{-8} \\ \\ x=\frac{-40\pm\sqrt{2000}}{-8} \\ \\ x_1=\frac{-40+\sqrt{2000}}{-8}\approx-0.6 \\ \\ x_2=\frac{-40-\sqrt{2000}}{-8}\approx10.6 \end{gathered}$

x-intercepts: (-0.6, 0) and (10.6, 0)

Graph: In green the axis of symmetry

Domain of a funtion is the set of x-values for which the function is defined

Range of a function is the set of values the function takes

Domain: All real numbers (-∞,∞)Range: From -infionite to 125 (-∞,125]In the given situation the vertex states the maximum height of the rocket (125) and the time it takes to reach that maximum (5 seconds)

The solutionsrepresents the moment (time) the rocket is in the ground; as it starts at a height of 25 (y-intercept) the negative solution doesn't have sence in the situation;

the solution (10.6, 0) means that after 10.6 seconds of the launch of the rocket it hits the ground