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**Describing the graph of a Parabola**

There are four different ways to describe the graph of a parabola: It intersects the x-axis twice. It is tangent to the x-axis. (It only intersects it once) It lies entirely above the x-axis. It lies entirely below the x-axis. We are going to explore all of these ways by looking at the discriminant and the coefficient of the squared term.

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**We need to be able to determine if a parabola opens up or down**

We need to be able to determine if a parabola opens up or down. To do that, we look at the coefficient of the squared term. Now type the following equation into y= on your calculator. Type in the following equation into y= on your calculator. What did we learn? If the squared term is positive, it opens up. If it is negative, it opens down!

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**We need to be able to determine what makes a parabola touch the x-axis and lie above or below it:**

The discriminant is a positive, non-perfect square. Therefore the roots are: 1. Real 2. Irrational 3. Unequal If the roots are real, the parabola touches the x-axis: Ex: We would describe this graph as touching the x-axis twice!

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**If the roots are imaginary, the parabola doesn’t touch the x-axis:**

Opens down The discriminant is negative. Therefore the roots are: 1. Imaginary We would describe this graph as follows: It lies entirely below the x-axis. If the roots are imaginary and the coefficient of the squared term is negative, it lies entirely below the x-axis. If the roots are imaginary and the squared term is positive, it lies entirely above the x-axis.

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Page 2 Now lets determine what the graph looks like. #16: #13: Parabola opens up Parabola opens up The discriminant is a positive, non-perfect square. Therefore the roots are: 1. Real 2. Irrational 3. Unequal The discriminant is a positive, perfect square. Therefore the roots are: 1. Real 2. Rational 3. Unequal Since the roots are unequal and real, the graph intersects the x-axis twice! Since the roots are unequal and real, the graph intersects the x-axis twice!

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Page 2 #19: #20: Parabola opens up Parabola opens down The discriminant is zero. Therefore the roots are: 1. Real 2. Rational 3. Equal The discriminant is negative. Therefore the roots are: 1. Imaginary Since the roots are equal the parabola is tangent to the x-axis. Since the roots are imaginary and the parabola opens down, then the parabola lies entirely below the x-axis.

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Page 2 #34: The roots of are equal when k is: Remember, the roots are equal when the discriminant is 0.

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Page 2 #36: The roots of are imaginary when b is: Remember, the roots are imaginary when the discriminant is negative. < 0 means negative! Since this is multiple choice, sub in values from choices until one works!

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Page 2 #38: If the graph of is tangent to the x-axis, then the roots of are: If a graph is tangent to the x-axis, it only touches the x-axis__________. ONCE The roots are: Real Rational Equal

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Page 2 #36: The roots of are equal when b is: Remember, the roots are equal when the discriminant is zero. = 0 means equal roots.

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**Find the largest integral value of k such that the roots of the given equation are real.**

means real.

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Homework Page 2 #15,18,21,35,41, 43,46

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