Task 1

Name: Andrew

Question: What can you tell about the graph of a quadratic function based on the roots you get from solving the quadratic formula for that equation?

Answer:  The quadratic formula can tell you how many roots or X-intercepts you can expect with a quadratic equation.  Using the discriminant, you can figure out the types of roots you will get.  The discriminant is what is found under the square root in the quadratic formula.  

                      in the formula

If the discriminant is a positive number, you will get 2 real roots because you will be adding or subtracting a positive root. 

If the discriminant is equal to zero, you will end up with 2 roots that are equal to each other because adding and subtracting zero give you the same value. 

If the discriminant is a negative number, you will get 2 non-real roots.  The square root of a negative number is a non-real number.  this will give you non-real or complex roots.

Example:

In this equation:  OR

Knowing that, we know that this equation will have 2 real roots.

By using the quadratic formula, we found that the roots are 3 and -1, both are real roots.  This proves the statement that when the discriminant is positive, there will be 2 real roots.

Task 2

The question: What can you tell about the range of an exponential function by looking at it in transformational form.

The answer: The range is determined by the y values.  The first thing to look for is the vertical translation.  In transformational form it would appear like this with VT being the vertical translation.

This value is the horizontal asymtote and the value for which all values of y will be either greater than or less then.  Now look for the vertical stretch.  If this value is positive, then the range will go from the horizontal asymtote to positive infinity {y>VT, yЄR} OR yЄ[VT,∞).  If the horizontal stretch is negative, then there is a reflection in x, and all y values will be less than the horizontal asymtote {y<VT, yЄR} OR yЄ(-∞,VT].

This is a graph of  Or in transformational form .  The function y=2 in red represents the horizontal asymtote.

The transformations:  Vertical Translation: 2

                                   Horizontal translation: 3

                                   Vertical stretch: 3

                                   Horizontal stretch: 1

The range: {y>2, yЄR}

The horizontal asymtote: y=2

the y-intercept: (0,2.375)