(b)
IM 2 Determining the Vertex Form of a Quadratic Function
Ex. Determine the vertex form equation y = a x − k
2
+ h of each quadratic function graphed below by considering the vertical stretch factor and both horizontal and vertical shifts.
(a)
( )
(d)
(f)
(c)
(a) _______________________________
(e)
(b) _______________________________
(c) _______________________________ (d) _______________________________
(e) _______________________________ (f) _______________________________
*** When you are finished, check all your answers on DESMOS. J
It is pretty easy to graph a parabola in vertex form y =
( ) 2 + h -- all you have to do is locate the coordinates of the vertex then apply the vertical stretch factor to find some key points on the graph.
However, what if you are given a quadratic function in standard form
y
= ax 2 + bx + c ? Not so easy!
Ex. Convert the standard form equation
y
= x 2 − 4
State the coordinates of the vertex in the form x + 7
V k ,
into vertex form by
( )
. completing the square .

Ex. It's your turn to do these two now...
(a) y
= x
2 −
10 x
+
19 (b) y
= x
2 +
5 x
+
12
Ex. Convert each quadratic function into vertex form by completing the square , and then draw its graph on the set of axes given. These ones are more difficult than the last three -- why?!
(a)
y
= − 2 x 2 − 20 x − 53 (b) y =
1
2 x 2 + 6 x + 14

2.
3.
Exercises
1. Convert each quadratic function to vertex form by completing the square. Then draw its graph on graph paper by using your knowledge of the vertical stretch and the two translations.
(a)
( )
= x 2 + 18 x + 85 (b)
( )
= − x 2 + 2 x − 5
(c)
( )
= 5 x 2 − 50 x + 123 (d)
( )
= −
1
4 x 2 − 2 x − 1
(a) On a fresh DESMOS screen, draw the graphs of
f using the standard form equations given.
,
g
, h , and i from Question #1 by
(b) Now graph your vertex from equations from Question #1. They should all match part (a)!
If any do not match, go back to your algebraic work in Question #1 and find your mistake.
When a ball is thrown through the air, the height of the object above the ground follows parabolic path -- we have to ignore pesky factors such as air resistance, but it’s essentially a parabola!
Let’s say that t seconds after a ball is thrown, its height h above the ground in meters is given by the formula h = − 5 t 2 + 20 t + 1.6
.
(a) How high above the ground was the ball when it was thrown? Why?
(b) Use your GDC to help draw a reasonable sketch on the set of axes showing the height h of the ball above the ground t seconds after it was thrown. Your GDC can easily find the coordinates of the vertex, but do you know how to ask it do so? If not, ask me! Think about why you only draw the graph in Quadrant 1!
(c) Now, you may have noticed that the title of this course is IM 2 not GDC 2! You still need to be able to use technology when appropriate, but you also have to be able to demonstrate a solid understanding of how to use algebraic techniques to solve problems involving quadratic equations. So, even though you already know the answer to the next question from part (b)…
Use algebra to find the maximum height the ball reaches by completing the square on the equation to determine the vertex. Write a good statement including units, and also state the time at which the ball reaches it maximum height.
(d) By using the vertex form of the equation, determine how long it takes (to the nearest tenth of a second) for the ball to hit the ground. What intercept are you finding? Verify your answer by using the graph on your GDC to find the zero.