Lesson Plan
Lecture Edition
Quadratics: vertex form
Objectives
Students will:
 Graph quadratic equations.
 Find the vertex of quadratic functions.
 Determine if the graph opens upward or downward.
 Convert quadratic equations to vertex form from polynomial form.
Prerequisite Knowledge
Students are able to:
 Plot points on the coordinate plane.
 Graph linear functions.
 Solve equations for a given variable.
 Shift and reflect functions.
Resources
 This lesson assumes that your classroom has only one computer, from
which you can lecture. If your classroom has enough computers for all
your students, either working individually or in small groups, see the
lab version of this lesson.
 Rulers, pencil, paper, a coin or tennis ball
 Access to http://www.explorelearning.com/
 Copies of the worksheet for each student (optional)
Lesson Preparation
Before conducting this lesson, be sure to read through it thoroughly, and
familiarize yourself with the Quadratics: vertex form activity at
ExploreLearning.com and Activity: Golf Range on ExploreLearning.com. You
may want to bookmark the activity page for your students. If you like make,
copies of the worksheet for each student.
Lesson
Motivation:
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Go to Activity: Golf Range on ExploreLearning.com. This activity allows users to
launch a golf ball at different angles and velocities.
Have students try to get a hole-in one. They will need to use the slide bars to
select a launch angle and a launch velocity. Once these are set, they need to
click the launch button to fire the golf ball. To see the path of the ball, students
can select the “trails” button. Ask students what is the general shape of the path
of the golf ball. The graphs should look something like the one below.
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B
Height of ball
C
A
Distance from tee
Ask students to give reasons why the graph had this shape. Ask students where
the height of the ball increased most rapidly. Ask students what happened at
point B. Ask students to compare the velocity of the ball at points A and C. Ask
students if a linear equation could be used to model the flight of the golf ball.
Students should see that linear equations do not model this graph well. Hence a
new type of function is needed to model the graph. This should lead nicely to a
discussion about quadratic equations.
The “Quadratics: vertex form” activity
To explore quadratic functions go to the Quadratics: vertex form activity at
ExploreLearning.com.
The meaning of ‘a’
Grab the ‘a’ slide bar and slide it left and right. Ask students what happens when
the sign of ‘a’ changes. Ask students what happens when ‘a’ is 0. Have students
make conjectures relating the sign of ‘a’ to the graph.
Now have students notice what happens to the graph as ‘a’ approaches 0 and as
‘a’ moves away from 0. Students should notice that the graph gets “steeper” as ‘a’
gets farther away from 0, and “wider” as ‘a’ approaches 0.
The meaning of ‘h’
Select the “show vertex/intercept data” box, then grab the ‘h’ slide bar and move
it left and right. Ask students what happens to the graph as you move the slide
bar to the right. Ask students what happens to the graph as you move the slide
bar to the left. Have students notice the x coordinate of the vertex as ‘h’ changes
value. Have students make conjectures relating ‘h’ to the vertex of a quadratic.
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The meaning of ‘k’
Grab the ‘k’ slide bar and move it left and right. Ask students what happens to
the graph as you move the slide bar to the right. Ask students what happens to
the graph as you move the slide bar to the left. Have students notice the y
coordinate of the vertex as ‘k’ changes value. Have students make conjectures
relating ‘k’ to the vertex of a quadratic.
Putting it all together
Have students predict how the graph of y = -3(x + 2)2 – 6 would look. Ask them
what quadrant the vertex would be in. Ask them the coordinates of the vertex.
Ask them if the graph would open up or down. Go over the student responses.
Type in the values for ‘a’, ‘h’, and ‘k’ to the right of the appropriate slide bars.
The graph will appear and the students can check their answers.
Have students graph several quadratic functions.
Ask students how the graph of y = -3(x - 1)2 + 3 differs from the graph of
y = -3(x + 2)2 – 6. Students should see that the vertex shifts to the right 3 units
and shifts up 9 units.
Graph y = -3(x - 1)2 + 3 and have students check their answers.
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Suppose the function y = 2x2 passes through the point (2, v) and the function
y = 8x2 passes through the point (2, w). Have students make conjectures relating
‘v’ to ‘w’.
To test their conjectures you can graph y = 2x2 and y = 8x2 and select the
‘Calculate data clipboard’ at the bottom of the screen for each function. They can
compare y values for a variety of x values from this table.
y = 2x2
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y = 8x2
Have students make conjectures relating ‘a’, ‘h’, and ‘k’ to the number of xintercepts the graph of y = a(x – h)2 + k has. Students should be able to see that
quadratics have 0, 1, or 2 x-intercepts depending on the location of the vertex
and the sign of ‘a’. Graph several examples of each type of quadratic to test their
conjectures.
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Maximum and minimum values
Graph y = -2(x2 – 4) + 6.
Ask students to find the maximum value that y attains for this function. Ask
students where this maximum value for y occurs. Ask students when a quadratic
function would have a minimum value for y. Have students make conjectures
about the maximum and minimum values of a quadratic function. Give students
the following application problem.
A farmer has 100 feet of fence in which to build a rectangular pen for his
chickens. The farmer would like to build the pen in such a way as to maximize
the area of the pen. The area function for this pen is defined as
y = -(x – 25) 2 + 625. Find the dimension of the pen when the area is maximized.
Have students explain their methods of solving the problem.
Converting quadratic equations from polynomial form to vertex form.
Have students write the equation y = 2x2 – 12x + 22 on their paper. Ask students
the coordinates of the vertex for this quadratic function. Ask students why it is
difficult to determine the coordinates of the vertex. Students should see that the
equation is not in the proper form to readily determine the vertex. Go over the
steps involved in converting quadratic equations from polynomial form to vertex
form. This method is called “completing the square.”
Step 1. Group the ‘x’ terms together
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y = (2x2 – 12x) + 22
Step 2. Factor out the leading coefficient of the ‘x2‘ term
y = 2(x2 – 6x) + 22
Step 3. Find ½ of the ‘x’ term then square the answer.
½(-6) = -3
(-3)2 = 9
Step 4. Insert the answer from step 3 into the parenthesis as a constant
term.
2(x2 – 6x + 9)
Step 5. Multiply the coefficient of the parenthesis by the constant term to
determine the amount of change in value from the original grouping.
2(x2 – 6x + 9)
2(9) = 18
Step 6. Add or subtract an equivalent amount from the constant term on
the outside of the parenthesis to keep the equality of the equation.
y = 2(x2 – 6x + 9) + 22 - 18
y = 2(x2 – 6x + 9) + 4
Students should now readily identify the vertex of this quadratic. Have students
convert several more quadratics to vertex form.
Conclusion
Quadratic equations in the form y = a(x – h)2 + k have a vertex at (h, k). If a > 0
the graph opens up, and when a < 0 the graph opens down. The magnitude of ‘a’
also determines the “steepness” of the graph. The graphs of quadratic functions
can have 0, 1, or 2 x-intercepts.
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