Back to last slideMain Menu Graphing, Max/Min, and Solving By Mrs. Sexton Calculator Tips.

Back to last slideMain Menu

Graphing, Max/Min, and Solving

By Mrs. Sexton

Calculator Tips


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Introduction to Quadratic Equations

Examples of Quadratic Equations

Quadratic Equation Facts

Calculator Tips

Quadratic Equation Practice Problems

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Introduction to Quadratic Equations

Quadratic Equations can always written in one of the two forms below.

cbxaxy 2

cbxaxxf 2)(

OR

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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.

Intro (cont.)

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Examples of Quadratic Equations

2xy

cbxaxy 2

0,0,1 cba

xxy 53 2 0,5,3 cba

1025.0 2 xy 10,0,25.0 cba

842 xxy 8,4,1 cba

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56231 xxy 5,6,3

1 cba


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.)

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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.

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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.

Axis of symmetryVertex

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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

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Quadratic Equation Practice Problems


232 xxy•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.

5.1x 25.0y

Vertex: (1.5, -0.25)

5.1xAxis of Symmetry:

Minimum value of the function: y = -0.25

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232 xxy

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•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.

1x 0y

Solution: x=1

•Repeat this process for the other solution. You end up with x = 2.


235.0 2 xxy•Type this equation into the Y1= in your calculator.



•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.

3x 5.2y

Vertex: (3, 2.5)

3xAxis of Symmetry:

Maximum value of the function: y = 2.5

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235.0 2 xxy

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•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.


734.0x 0y

Solution: x=0.734

•Repeat this process for the other solution. You end up with x = 5.24.


325.0 2 xy•Type this equation into the Y1= in your calculator.



•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.

0x 3y

Vertex: (0, 3)

0xAxis of Symmetry:

Maximum value of the function: y = 3

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•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.


46.3x 0y

Solution: x=-3.46

•Repeat this process for the other solution. You end up with x = 3.46.

325.0 2 xy


346 2 xxy•Type this equation into the Y1= in your calculator.


•Look at the graph. Find the top or the bottom of the graph—the “vertex.” Is it a maximum or a minimum? 33.0x 33.2y

Vertex: (0.33, 2.33)

33.0xAxis of Symmetry:

Minimum value of the function: y = 2.33

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•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.



346 2 xxy

????Since the graph of the function does not cross the x-axis, there are

NO REAL SOLUTIONS.


Calculator Tips (Page 1)

• 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.

Entering and graphing the quadratic equation

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• 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.

Finding the minimum value of a function

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• 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.

Finding the maximum value of a function

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• 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).

Finding the roots or solutions of a quadratic equation

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• 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.

Finding the axis of symmetry of a quadratic equation

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