1. Linear vs Quadratic Graphs MAP4C

2. Degrees  Recall:  The degree of a polynomial is given by the highest power (exponent) in any term of the polynomial  A linear relation is a straight-line graph with a general form: y = mx + b. The algebraic expression involved is a polynomial of degree 1. Ex: y = 3x - 5  A quadratic function is a parabola with a general form: y=a(x - h) 2 + k. It is a polynomial with a degree of 2.  Ex: f(x) = 2x 2 + 3x – 5

3. Using Data Tables and Graphs  Data Table of Distance and time for a moving object:

4. Data Tables and Graphs  We can see that the first differences are constant, so it is a linear relation.  Plot this graph.  A scatter-plot yields a linear line of best-fit.  The equation can be found from the slope. m = rise/run =12 m/s and y-intercept b = 0.  The general linear equation y = mx + b can be stated as y = 12 x Degree = 1

5. Using Data Tables and Graphs  Data Table of Distance and time for a moving object: Time (s)Distance (m) First DifferencesSecond Differences 0 0 1 3 2 12 3 27 4 48 5 75

6. Data Tables and Graphs  2 nd differences are the same, so this motion can be represented by a polynomial of degree 2 and is a quadratic function.  A plot of this data would yield a parabola, starting at the origin and opening upwards.  The object in this data is accelerating. y = 3x 2 is the equation. The degree is 2

7. Questions  What is the relationship between the degree of a function and the differences in a table of values?  How can you tell if a function is linear or quadratic from a:  Table of values?  Graph?  Equation?

8. Quadratic Properties  All quadratics have a degree of 2 y = a(x – h) 2 + k  They usually have a vertex (h, k) and open up or down  The vertex is a maximum if the quadratic opens down.  The vertex is a minimum if the quadratic opens up.  The x-intercepts are called the roots, or solutions for the equation of the quadratic yielding y = 0.  In a profit function, the x-intercepts indicate the break-even points (when profit is 0).  A vertical translation (k) shifts the quadratic up or down, altering the vertex location.

9. Text work  Page 463 #1-4, 6, 7, 12, 14