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The quadratic equation $h=\mathrm{-16}{t}^{2}+128t+32$ is used to find the height of a stone thrown upward from a height of 32 feet at a rate of 128 ft/sec. How long will it take for the stone to reach its maximum height? What is the maximum height? Round answers to the nearest tenth.
It will take 4 seconds to reach the maximum height of 288 feet.
A toy rocket shot upward from the ground at a rate of 208 ft/sec has the quadratic equation of $h=\mathrm{-16}{t}^{2}+208t$ . When will the rocket reach its maximum height? What will be the maximum height? Round answers to the nearest tenth.
It will take 6.5 seconds to reach the maximum height of 676 feet.
Access these online resources for additional instruction and practice graphing quadratic equations:
Recognize the Graph of a Quadratic Equation in Two Variables
In the following exercises, graph:
$y=\text{\u2212}{x}^{2}+1$
In the following exercises, determine if the parabola opens up or down.
$y=6{x}^{2}+2x+3$
$y=\mathrm{-9}{x}^{2}-24x-16$
Find the Axis of Symmetry and Vertex of a Parabola
In the following exercises, find ⓐ the axis of symmetry and ⓑ the vertex.
$y={x}^{2}+8x-1$
ⓐ $x=\mathrm{-4}$ ⓑ $\left(\mathrm{-4},\mathrm{-17}\right)$
$y={x}^{2}+10x+25$
$y=\mathrm{-2}{x}^{2}-8x-3$
Find the Intercepts of a Parabola
In the following exercises, find the x - and y -intercepts.
$y={x}^{2}+7x+6$
$y\text{:}\phantom{\rule{0.2em}{0ex}}(0,6);\phantom{\rule{0.2em}{0ex}}x\text{:}\phantom{\rule{0.2em}{0ex}}(\mathrm{-1},0),(\mathrm{-6},0)$
$y={x}^{2}+10x-11$
$y=\text{\u2212}{x}^{2}+8x-19$
$y\text{:}\phantom{\rule{0.2em}{0ex}}(0,19);\phantom{\rule{0.2em}{0ex}}x\text{:}\phantom{\rule{0.2em}{0ex}}\text{none}$
$y={x}^{2}+6x+13$
$y=4{x}^{2}-20x+25$
$y\text{:}\phantom{\rule{0.2em}{0ex}}(0,25);\phantom{\rule{0.2em}{0ex}}x\text{:}\phantom{\rule{0.2em}{0ex}}(\frac{5}{2},0)$
$y=\text{\u2212}{x}^{2}-14x-49$
Graph Quadratic Equations in Two Variables
In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry.
$y={x}^{2}+6x+5$
$y\text{:}\phantom{\rule{0.2em}{0ex}}(0,5);\phantom{\rule{0.2em}{0ex}}x\text{:}\phantom{\rule{0.2em}{0ex}}(\mathrm{-1},0),(\mathrm{-5},0);$
axis:
$x=\mathrm{-3};\phantom{\rule{0.2em}{0ex}}\text{vertex}\text{:}\phantom{\rule{0.2em}{0ex}}(\mathrm{-3},\mathrm{-4})$
$y={x}^{2}+4x-12$
$y={x}^{2}+4x+3$
$y\text{:}\phantom{\rule{0.2em}{0ex}}(0,3);x\text{:}\phantom{\rule{0.2em}{0ex}}(\mathrm{-1},0),(\mathrm{-3},0);$
axis:
$x=\mathrm{-2};\phantom{\rule{0.2em}{0ex}}\text{vertex:}\phantom{\rule{0.2em}{0ex}}(\mathrm{-2},\mathrm{-1})$
$y={x}^{2}-6x+8$
$y=9{x}^{2}+12x+4$
$y\text{:}\phantom{\rule{0.2em}{0ex}}(0,4)\phantom{\rule{0.2em}{0ex}}x\text{:}\phantom{\rule{0.2em}{0ex}}(-\frac{2}{3},0);$
axis:
$x=-\frac{2}{3};\phantom{\rule{0.2em}{0ex}}\text{vertex:}\phantom{\rule{0.2em}{0ex}}(-\frac{2}{3},0)$
$y=\text{\u2212}{x}^{2}+8x-16$
$y=\text{\u2212}{x}^{2}+2x-7$
$y\text{:}\phantom{\rule{0.2em}{0ex}}(0,\mathrm{-7});x\text{:}\phantom{\rule{0.2em}{0ex}}\text{none};$
axis:
$x=1;\text{vertex}\text{:}\phantom{\rule{0.2em}{0ex}}(1,\mathrm{-6})$
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