10.1 Exploring Quadratic Graphs
Lets get out our graphing calculators . . .
• start with the quadratic function x
we can write y = x OR f(x) = x
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What happens when we change the coefficient of the quadratic (a) . . .
• like 2x
we can write y = 2x OR f(x) = 2x
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• Which function above represents y = x and which function represents y = 2x ?
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• What would happen if we changed the quadratic coefficient to 10, to 20, to 100? (Test your theories using your graphing calculator.)
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• What would happen if we changed the quadratic coefficient to ­1, to ­2, to ­10? (Test your theories using your graphing calculator.)
y = -x2
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• What would happen if we changed the quadratic coefficient to 0.5, to 0.1, to 0.01? (Test your theories using your graphing calculator.)
y = 0.5x2
• Which graph represents y = x and y = 0.5x ?
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• How does the coefficient of the quadratic term affect the graph?
If a > 1, the graph becomes steeper and more narrow
If 0 < a < 1, the graph becomes wider
If ­1 < a < 0, the graph reflects over the x­axis and becomes more narrow
If a < ­1, the graph reflects over the y­axis and becomes wider
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Identify the vertex of each graph. Determine whether the vertex is a minimum or maximum. State the axis of symmetry.
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Order each group of quadratic function from widest to narrowest graphs.
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Make a table:
Graph the points:
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You try:
f(x) = 4x2
x
4x2 f(x) 0 4(0)(0) 0 1 4(1)(1) 4 2 4(2)(2) 16 9
You try:
f(x) = -3x2
x
­3x2 f(x) 0 ­3(0)(0) 0 1 ­3(1)(1) ­3 2 ­3(2)(2) ­12 10
Notice all of the quadratic function we have looked at have a constant of _______
• What would happen if we changed the constant? (Test your theories using your graphing calculator.)
• Let's try y = x + 1
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y = x2+1
What happened? What would happen if we made the constant 5, or 8? (Test your theories using your graphing calculator.)
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• What would happen if we made the constant negative? (Test your theories using your graphing calculator.)
• Let's try y = x ­ 4
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y = x2 - 4
What happened? What would happen if we made the constant ­5, or ­8? (Test your theories using your graphing calculator.)
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Let's graph a few:
f(x) = x2-3
x
x2 ­ 3
f(x) 0 (0)(0) ­ 3 ­3 1 (1)(1) ­ 3 ­2 2 (2)(2) ­ 3 1 13
Let's graph a few:
f(x) = x2+ 7
x
x2 + 7
f(x) 0 (0)(0) + 7 7 1 (1)(1) + 7 8 2 (2)(2) + 7 11 14
Let's graph a few:
f(x) = -2x2+ 3
x
­2x2 + 3
f(x) 0 ­2(0)(0) + 3 3 1 ­2(1)(1) + 3 1 2 ­2(2)(2) + 3 ­5 15
Let's graph a few:
f(x) = 0.25x2-3
OR
x
0.25x2 ­ 3
f(x) x
0 0 1 4 2 8 0.25x2 ­ 3
f(x) Why did I use 0, 4, and 8?
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