Aim #61: How do we graph quadratic equations from the vertex form?
Homework: handout
Do Now: a. Graph each of the following quadratic equations (you may use a
calculator to help you find points).
y = x2
vertex:
y = (x - 2)2
vertex:
y = (x + 2)2
vertex:
b. How are the coordinates of the vertex for each graph similar? How are they
different?
1. a. The graph of y = x2 is shown below. Consider the graph of y = (x − 4)2. Where
would you expect this graph to be in relation to y = x2? Sketch a graph of
y = (x - 4)2.
b. What is the vertex of
y = (x - 4)2?
c. Graph y = (x - 4)2 + 1 on the axes above. How does this graph relate to the
graph of y = (x - 4)2? What is the vertex of this quadratic?
2. a. Graph y = (x + 3) 2 - 5 on the axes.
b. Describe how this quadratic equation
relates to y = x 2 , the parent function.
c. What is the vertex of y = (x + 3)
2
- 5?
d. How does the vertex relate to the
equation?
3. Without graphing, state the vertex for each of the following quadratic
equations.
a. y = (x - 5)2 + 3
b. y = x 2 - 25
c. y = (x + 4) 2
4. Write a quadratic equation whose graph will have the given vertex.
a. (-2, 6)
b. (1.9, -4)
c. (0, 100)
d. (h, k)
5. a. The graph of y = (x - 3) 2 + 1 is shown below. Graph the following
quadratic equations on the same axes.
y = 2(x - 3)2 + 1
y=
1
(x - 3)2 + 1
4
y = -4(x - 3)2 + 1
b. What do they all have in common?
The vertex form of a quadratic equation is y = a(x - h)2 + k, where (h, k) is the
vertex.
Compared to when a = 1,
the graph is shrunk vertically when - 1 < a < 1 (the graph gets wider)
the graph is stretched vertically when a < -1 or a > 1 (the graph gets more narrow)
the graphs opens up when a is positive
the graph opens down when a is negative
6. Compare the graphs of the function f(x) = -2(x + 3) 2 + 2 and g(x) = 5(x + 3)2 + 2.
7. Write two different equations representing quadratic functions whose graphs
have vertices at (4.5, -8).
A quadratic equation in standard form can be converted into an equation in
vertex form by using completing the square.
8. Convert to vertex form and then state the vertex:
y = x2 + 4x - 5
Let's sum it up!!!
The parent function of all quadratic equations is y = x2.
The vertex form of a quadratic equation is y = a(x - h)2 + k, where (h,k)
are the coordinates of the vertex. The a value gives us information about
stretching, shrinking, open up or open down.
A quadratic equation in standard form can be converted into vertex form by using
completing the square.