Graphing, Max/Min, and Solving
By Mrs. Sexton
Calculator Tips
Main Menu
Back to last slide
Main Menu
Introduction to Quadratic Equations
Examples of Quadratic Equations
Quadratic Equation Facts
Quadratic Equation Practice Problems
Calculator Tips
Calculator Tips
Main Menu
Back to last slide
Introduction to Quadratic
Equations
Quadratic Equations can always written in one of the two forms
below.
y  ax  bx  c
2
OR
f ( x)  ax  bx  c
2
Calculator Tips
Main Menu
Back to last slide
Intro (cont.)
A quadratic function always has a degree of 2. That
means that there’s always an x2 in the equation and
never any higher power of x. There may or may not be
a bx term and there may or may not be a c.
NOTE: Quadratic equations never have more than one
independent variable in the equation. The variable y is
dependent, and f(x) means that the equation is a function
of the independent variable x.
Calculator Tips
Main Menu
Back to last slide
Examples of Quadratic Equations
y  ax  bx  c
2
yx
2
a  1, b  0, c  0
y  3x  5 x
a  3, b  5, c  0
2
y  0.25x 10
a  0.25, b  0, c  10
y  x2  4x  8
a  1, b  4, c  8
y  x  6x  5
a  13 , b  6, c  5
2
1
3
2
Calculator Tips
Main Menu
Back to last slide
Quadratic Equation Facts
• The coefficient a in front of the x2 term tells
you if the graph opens up or down. (If a is
positive, the graph opens up. If a is a
negative number, the graph opens down.)
• The a also gives you an idea if the graph is
narrow or wide. (If a is a fraction, the
graph is wide. If it’s a number bigger than
1 or less than –1, the graph is narrow.)
Calculator Tips
Main Menu
Back to last slide
Facts (cont.)
• The point where the graph reaches its
highest point or its lowest point is called the
vertex. This point is an ordered pair.
• The vertex can be either a maximum or a
minimum. If the graph opens up, the
function has a minimum.
If it opens
down, the function has a maximum.
Calculator Tips
Main Menu
Back to last slide
Facts (cont.)
• Quadratic functions are always symmetric
about a vertical line. This line is called the
axis of symmetry. The vertex is always on
the axis of symmetry, and this line acts like
a mirror reflecting the graph.
Vertex
Calculator Tips
Axis of symmetry
Main Menu
Back to last slide
Facts (last page )
• The roots, solutions, zeroes, or x-intercepts
of a quadratic equation all mean the same
thing. This is where the graph of the
function crosses the x-axis. The y-value is
always zero for these points.
Roots, solutions, zeroes, x-intercepts
Calculator Tips
Main Menu
Back to last slide
Quadratic Equation Practice
Problems
Calculator Tips
Main Menu
Back to last slide
y  x  3x  2
2
•Type this equation into the Y1= in your
calculator.
•Graph it. Adjust your window if needed.
•Look at the graph. Find the top or the
bottom of the graph—the “vertex.” Is it a
maximum or a minimum?
•Since this is the absolute lowest point
that this graph can go, it is a
minimum. Use your calculator to find
the vertex, the axis of symmetry, and
the minimum value of the function.
Calculator Tips
Main Menu
x  1 .5
y  0.25
Vertex: (1.5, -0.25)
Axis of Symmetry:
x  1 .5
Minimum value of the
function: y = -0.25
Back to last slide
y  x  3x  2
2
•Now, find the solutions of this
quadratic function on your calculator.
(Remember that the solutions may also
be called roots, zeroes, or x-intercepts.)
•Choose an x-value to the left of this root
(say 0, for example) for your Left Bound.
•Choose an x-value to the right of this
root (say 1.5, for example) for your Right
Bound.
•Press “Enter” to Guess.
Calculator Tips
x 1
y0
Solution: x=1
•Repeat this process for the other
solution. You end up with x = 2.
Main Menu
Back to last slide
y  0.5x  3x  2
2
•Type this equation into the Y1= in your
calculator.
•Graph it. Adjust your window if needed.
•Look at the graph. Find the top or the
bottom of the graph—the “vertex.” Is it a
maximum or a minimum?
•Since this is the absolute highest
point that this graph can go, it is a
maximum. Use your calculator to
find the vertex, the axis of symmetry,
and the maximum value of the
function.
Calculator Tips
Main Menu
x3
y  2.5
Vertex: (3, 2.5)
Axis of Symmetry:
x3
Maximum value of the
function: y = 2.5
Back to last slide
y  0.5x  3x  2
2
•Now, find the solutions of this
quadratic function on your calculator.
(Remember that the solutions may also
be called roots, zeroes, or x-intercepts.)
•Choose an x-value to the left of this root
(say 0, for example) for your Left Bound.
•Choose an x-value to the right of this
root (say 1.5, for example) for your Right
Bound.
•Press “Enter” to Guess.
Calculator Tips
x  0.734
y0
Solution: x=0.734
•Repeat this process for the other
solution. You end up with x = 5.24.
Main Menu
Back to last slide
y  0.25x  3
2
•Type this equation into the Y1= in your
calculator.
•Graph it. Adjust your window if needed.
•Look at the graph. Find the top or the
bottom of the graph—the “vertex.” Is it a
maximum or a minimum?
•Since this is the absolute highest
point that this graph can go, it is a
maximum. Use your calculator to
find the vertex, the axis of symmetry,
and the maximum value of the
function.
Calculator Tips
Main Menu
x0
y3
Vertex: (0, 3)
Axis of Symmetry:
x0
Maximum value of the
function: y = 3
Back to last slide
y  0.25x  3
2
•Now, find the solutions of this
quadratic function on your calculator.
(Remember that the solutions may also
be called roots, zeroes, or x-intercepts.)
•Choose an x-value to the left of this root
(say -4, for example) for your Left Bound.
•Choose an x-value to the right of this
root (say -2, for example) for your Right
Bound.
•Press “Enter” to Guess.
Calculator Tips
x  3.46
y0
Solution: x=-3.46
•Repeat this process for the other
solution. You end up with x = 3.46.
Main Menu
Back to last slide
y  6x  4x  3
2
•Type this equation into the Y1= in your
calculator.
•Graph it. Adjust your window if needed.
•Look at the graph. Find the top or the
bottom of the graph—the “vertex.” Is it a
maximum or a minimum?
•Since this is the absolute lowest
point that this graph can go, it is a
minimum. Use your calculator to
find the vertex, the axis of symmetry,
and the minimum value of the
function.
Calculator Tips
Main Menu
x  0.33
y  2.33
Vertex: (0.33, 2.33)
Axis of Symmetry:
x  0.33
Minimum value of the
function: y = 2.33
Back to last slide
y  6x  4x  3
2
•Now, find the solutions of this
quadratic function on your calculator.
(Remember that the solutions may also
be called roots, zeroes, or x-intercepts.)
Since the graph of the function
does not cross the x-axis, there
are
????
NO REAL SOLUTIONS.
Calculator Tips
Main Menu
Back to last slide
Calculator Tips (Page 1)
Entering and graphing the quadratic equation
• Type your quadratic equation into Y1=
• Graph the equation. Press “Graph.”
• Adjust the window to see the vertex and the roots of the
equation.
– Use the Zoom Standard (6) button or the Zoom Fit (0)
button to see more of the graph. You may also press
“Window” and change the Xmin, Xmax, Ymin, and
Ymax to see the graph.
Calculator Tips
Main Menu
Back to last slide
Calculator Tips (Page 2)
Finding the minimum value of a function
• If the graph opens up, find the minimum value.
– Press 2nd, Trace (Calc), Minimum (3)
– Move the cursor (using the arrow keys) to the left of the
minimum. Press “Enter” for the “Left Bound” when you get to
the point you want.
– Move the cursor (using the arrow keys) to the right of the
minimum. Press “Enter” for the “Right Bound” when you get to
the point you want.
– Press “Enter” again for the calculator to “Guess.”
– The x and y values that appear at the bottom of the graph are the
x and y values of the vertex (x, y). The minimum value is the y
value at this point.
Calculator Tips
Main Menu
Back to last slide
Calculator Tips (Page 3)
Finding the maximum value of a function
• If the graph opens down, find the maximum value.
– Press 2nd, Trace (Calc), Maximum (4)
– Move the cursor (using the arrow keys) to the left of the maximum.
Press “Enter” for the “Left Bound” when you get to the point you
want.
– Move the cursor (using the arrow keys) to the right of the maximum.
Press “Enter” for the “Right Bound” when you get to the point you
want.
– Press “Enter” again for the calculator to “Guess.”
– The x and y values that appear at the bottom of the graph are the x and
y values of the vertex (x, y). The maximum value is the y value at this
point.
Calculator Tips
Main Menu
Back to last slide
Calculator Tips (Page 4)
Finding the roots or solutions of a quadratic equation
• To find the roots, solutions, zeroes, or x-intercepts of the equation
– Make sure you can see the root(s) in the viewing window. If not, adjust
your window.
– Press 2nd, Trace (Calc), Zero (2)
– Select an x-value to the left of the root you want to find. Press “Enter”
for the “Left Bound.”
– Select an x-value to the right of the root you want to find. Press
“Enter” for the “Right Bound.”
– Press “Enter” again to guess.
– The x-value that appears in the window is the root. The y-value should
be 0 (or very close to 0).
Calculator Tips
Main Menu
Back to last slide
Calculator Tips (Page 5)
Finding the axis of symmetry of a quadratic equation
• Find the maximum or the minimum using the instructions on the preceding
pages. The axis of symmetry is the equation for x = that appears on the
screen for the maximum or the minimum.
Calculator Tips
Main Menu
Back to last slide
Main Menu
Back to last slide