Write the equation of a parabola with a vertex of (2, -1) and which opens downward, algebra homework help

Write the equation of a parabola with a vertex of (2, -1) and which opens downward.
1. List three points you might test to find the solution of (x + 3)(x – 5) < 0. Explain why you chose these points.
2. Write a system of equations with one linear equation and one quadratic equation for which (2, 6) is a solution.
3. Use page 306 to Explain how the graph of y = x2 can be used to graph any quadratic function. Include a description of the effects produced by changing a, h, and k in the equation y = a(x – h)2 + k, and a comparison of the graph of y = x2 and the graph of y = a(x – h)2 + k using values of your own choosing for a, h, and k.
4. Use the information on page 314 to explain how you can find the time a trampolinist spends above a certain height. Include a quadratic inequality that describes the time the performer spends more than 10 feet above the ground, and two approaches to solving this quadratic inequality.ps: Find pages 306 and 314 attached below

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5-7
Analyzing Graphs of
Quadratic Functions
Concepts in Motion
Interactive Lab algebra2.com
Main Ideas
GET READY for the Lesson
A family of graphs is a group of graphs
that displays one or more similar
characteristics. The graph of y = x2 is
called the parent graph of the family of
quadratic functions.
• Analyze quadratic
functions of the form
y = a (x – h)2 + k.
• Write a quadratic
function in the form
y = a(x – 1)2 + k.
y = y2 + 2
LVIN
y=(x-3)
New Vocabulary
o
The graphs of other quadratic
functions such as y = x2 + 2 and
y = (x – 3)2 can be found by
transforming the graph of y = x2.
vertex form
Axis of
(0,0)
X=0
Analyze Quadratic Functions Each
function above can be written in the
form y = (x – h)2 + k, where (h, k) is
the vertex of the parabola and x = h is
its axis of symmetry. This is often
referred to as the vertex form of a
quadratic function.
Equation
y = x2 or
y = (x – 0)2 + 0
y = x2 + 2 or
y = (x – 02+2
y = (x – 3)2 or
y = (x – 3)2 + 0
(0,2)
X=0
(3,0)
X=3
Recall that a translation slides a figure
without changing its shape or size. As the values of h and k change, the
graph of y = a(x – h)2 + k is the graph of y = x2 translated:
• Th| units left if h is negative or h| units right if his positive, and
• |k|units up if k is positive or|k| units down if k is negative.
EXAMPLE Graph a Quadratic Equation in Vertex Form
Analyze y = (x + 2)2 + 1. Then draw its graph.
This function can be rewritten as y = [x – (-2)]2 + 1. Then h = -2
and k = 1. The vertex is at (h, k) or (-2, 1), and the axis of symmetry
is x = -2. The graph is the graph of y = x2 translated 2 units left and
1 unit up.
у
(0,5
2)
Now use this information to draw the graph.
(-4,5)
Step 1 Plot the vertex, (-2, 1).
Step 2 Draw the axis of symmetry, x = -2.
y= (x + 2)2 + 1
11
(-3, 27
Step 3 Use symmetry to complete the graph.
CHECK Your Progress
1. Analyze y = (x – 3)2 – 2. Then draw its graph.
(-2, 1)
o
X
5-8
Graphing and Solving
Quadratic Inequalities
Main Ideas
• Graph quadratic
inequalities in two
variables.
• Solve quadratic
inequalities in one
variable.
GET READY for the Lesson
Californian Jennifer Parilla is the only
athlete from the United States to
qualify for and compete in the
Olympic trampoline event.
New Vocabulary
quadratic inequality
Suppose the height h(t) in feet of a
trampolinist above the ground
during one bounce is modeled by
the quadratic function
h(t) = –16t+ 42+ + 3.75. We can
solve a quadratic inequality to
determine how long this performer
is more than a certain distance above
the ground.
Graph Quadratic Inequalities You can graph quadratic inequalities in
two variables using the same techniques you used to graph linear
inequalities in two variables.
Step 1 Graph the related quadratic
function, y = ax2 + bx + c. Decide
if the parabola should be solid
or dashed.
o
X
o
< or y Step 2 Test a point (x1, y1) inside the parabola. Check to see if this point is a solution of the inequality. M Y, 3 a(x,)2 + b(x,) + y! Step 3 If (x1, y1) is a solution, shade the region inside the parabola. If (x1, y1) is not a solution, shade the region outside the parabola. 0 X 0 (X1,Y) is a solution. (X1,Y,) is not a solution.