Question

for 9-10 describe each transformation of $f(x)=x^{2}$.9. $f(x)=(x-4)^{2}+8$10. $f(x)=-9(x+1)^{2}-1$11. explain how this graph transformed, answer should be an equation in vertex form $y=a(x-h)^{2}+k$12. the axis of symmetry is the y coordinate of the vertex.a. trueb. false13. when solving a systems of quadratics you look where the two graphs _____________a. goes upb. goes downc. intersectsd. turn14. range is the set of all the ___________ values.15. the city of clinton is planning their annual fireworks celebration for the $4^{th}$ of july. they have chosen explodarama to do the fireworks. y is the height in feet and x is the time in seconds.using the function find the information below.$y=-16x^{2}+150x+85$a. what is the max height the fireworks reach? (round to the nearest hundredth) ___________b. how long are they in the air? (round to the nearest tenth) ___________when would the fireworks be 261 feet off the ground? _____________

Explanation:

Step1: Identify base function $f(x)=x^2$

Base function: $f(x)=x^2$

Step2: Transform $f(x)$ for #9

Right 4, up 8: $f(x)=(x-4)^2+8$

Step3: Transform $f(x)$ for #10

Left 1, reflect over x-axis, vertical stretch by 9, down 1: $f(x)=-9(x+1)^2-1$

Step4: Analyze graph for #11

Vertex at $(0, 4)$, vertical stretch by 1, opens down: $y=-x^2+4$

Step5: Evaluate #12 statement

Axis of symmetry is $x=h$ (x-coordinate of vertex): false

Step6: Select #13 correct option

Systems of quadratics solve at intersections: intersects

Step7: Define range for #14

Range is all output (y) values: y

Step8: Find max height for #15a

Vertex x-coordinate: $x=-\frac{b}{2a}=-\frac{150}{2(-16)}=\frac{150}{32}=4.6875$
Substitute x into $Y$:
$Y=-16(4.6875)^2+150(4.6875)+85$
$Y=-16(21.97265625)+703.125+85$
$Y=-351.5625+703.125+85=436.5625\approx436.56$

Step9: Find air time for #15b

Set $Y=0$, solve $-16x^2+150x+85=0$
Quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
$a=-16, b=150, c=85$
$\Delta=150^2-4(-16)(85)=22500+5440=27940$
$\sqrt{27940}\approx167.15$
$x=\frac{-150\pm167.15}{-32}$
Positive solution: $x=\frac{-150+167.15}{-32}=\frac{17.15}{-32}$ (discard, negative)
$x=\frac{-150-167.15}{-32}=\frac{-317.15}{-32}\approx9.9$

Step10: Solve for $Y=261$ for #15c

Set $-16x^2+150x+85=261$
Rearrange: $-16x^2+150x-176=0$
Divide by -1: $16x^2-150x+176=0$
Quadratic formula: $x=\frac{150\pm\sqrt{150^2-4(16)(176)}}{2(16)}$
$\Delta=22500-11264=11236$
$\sqrt{11236}=106$
$x=\frac{150\pm106}{32}$
$x_1=\frac{150+106}{32}=\frac{256}{32}=8$
$x_2=\frac{150-106}{32}=\frac{44}{32}=1.375$

Answer:

1. The graph of $f(x)=x^2$ is shifted 4 units to the right and 8 units up.
2. The graph of $f(x)=x^2$ is shifted 1 unit to the left, reflected over the x-axis, vertically stretched by a factor of 9, and shifted 1 unit down.
3. $y=-x^2+4$
4. b. false
5. c. intersects
6. y

15a. 436.56 feet
15b. 9.9 seconds
15c. 1.375 seconds and 8 seconds