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Notes for 3.1 Quadratic Functions and Models
(pp. 294 - 303)
Topics: Quadratic Functions and Their Characteristics, Graphing, and Modeling
A function is a quadratic function if f ( ) x = ax^2 + bx + c , where a, b, and c are real numbers with
a โ 0.
- This is the algebraic form of the function.
I. Graphing Techniques ( pp. 295 โ 296, 299 โ 300) We begin with the basic quadratic function: Enter Y 1 = x^2
, wh
The axis of symmetry is x = 0, since that is the line of reflection from the left side of the graph to the right side of the graph. Notice the symmetry of the points in the TABLE, also.
The domain is ( โโ โ, )and the range is [ 0, โ ). (Mistake on the video graph.)
The quadratic function can also be written in the form ( )
2 f ( ) x = a x โ h + k , where h is the
horizontal translation (left or right) and k is the vertical translation (up or down).
Ex. Graph ( )
2 f ( ) x = x โ 2. State its domain and its range.
\
The domain is ( โโ โ, )and the range is [ 0, โ ).
Ex. Graph ( )
2 f ( ) x = x + 3 โ 4. State its domain and its range.
II. Completing the Square (Omit pp. 297 โ 298) We omit the completing the square technique here. Again, there are better ways to accomplish the same result.
III. Graphing a Quadratic Using the Algebraic Form of the Function (The Vertex Formula) (pp. 299 โ 300)
Recall: The quadratic function can be written as f ( ) x = ax^2 + bx + c (algebraic form) and also
2 f ( ) x = a x โ h + k (graphing form). The connection between the two forms is made by the
vertex ( h, k) , where 2
b h a
= and k = f h ( )(which is found by substituting the h value into the
quadratic.
Characteristics of a Quadratic Function:
- The graph is a _________ with the vertex (h, k) and the vertical line x = h as its axis of symmetry.
- It opens up if a > 0 (has a positive leading coefficient), or it opens down if a < 0 (has a negative leading coefficient).
- It is broader than the basic graph of x^2 if a < 1 (is a proper fraction), or
it is narrower than the basic graph of x^2 if a > 1 (is an improper fraction or whole number).
- The y-intercept is f (0)= c and is written as (0, c ).
- The x-intercept(s) are found from the quadratic formula: (^2 )
2
b b ac x a
= , if b^2^ โ 4 ac โฅ 0 (is positive or 0)
If b^2 โ 4 ac < 0 (is negative), then there are no x intercepts. (The graph floats above the x axis or remains below the x axis, so that it doesnโt cross there).
Ex. Find the axis of symmetry and the vertex of the parabola having the equation f ( ) x = 2 x^2 + 4 x + 5.
First, find the vertex:
b x a
Now, the y coordinate is found by substituting and finding f (-1) :
y = f( -1) = 2(-1)^2 + 4(-1) + 5 =
*Note that finding y is easily done by using the STO โ feature of the calculator.
II. Using Quadratics to Model Data
Ex. page 300, Example 5 a โ f a. Video b. The calculator WINDOW at the top sets the dimensions of the GRAPH screen. You can adjust the dimensions using WINDOW: XMIN is the left side of the x axis; XMAX is the right side of the x axis; XSCL is the scale that the calculator uses to write the hashmarks of the number line; YMIN sets the depth of the y axis; YMAX sets the height of the y axis; YSCL sets the distance between hashmarks on the y axis number line. [XMIN, XMAX, XSCL x (by) YMIN, YMAX, YSCL] is how some texts denote this information.
- ZOOM 6:Standard resets the WINDOW to [-10, 10, 1 x โ10, 10, 1] c. Video
