?v=cqj5Qvxd5MO Linear and Quadratic Functions and Modeling

A polynomial function is a function of the form: All of these coefficients are real numbers the exponent must be a nonnegative integer Remember integers are … –2, -1, 0, 1, 2 … (no decimals or fractions) so nonnegative integers would be 0, 1, 2 … The degree of the polynomial is the largest power on any x term in the polynomial.

Not a polynomial because of the square root since the power is NOT an integer Determine which of the following are polynomial functions. If the function is a polynomial, state its degree. A polynomial of degree 4. A polynomial of degree 0. We can write in an x 0 since this = 1. Not a polynomial because of the x in the denominator since the power is NOT nonnegative x 0

Graphs of polynomials are smooth and continuous. No sharp corners or cusps No gaps or holes---it can be drawn without lifting pencil from paper This IS the graph of a polynomial This IS NOT the graph of a polynomial

POLYNOMIAL FUNCTIONS OF NO AND LOW DEGREE NAME FORM DEGREE Zero function f(x) = 0 Undefined Constant function f(x) =a 0 Linear function f(x) ax + b 1 Quadratic function f(x) = ax² +bx + c 2 a ≠ 0

LINEAR EQUATIONS

In general, the equation of a horizontal line is y = b, where b is the y coordinate of any point on the line. If we look at any points on this line we see that they all have a y coordinate of 3 and the x coordinate varies. (-4, 3) (-1, 3) (2, 3) Let's choose the points (-4, 3) and (2, 3) and compute the slope. This makes sense because as you go from left to right on the line, you are not rising or falling (so zero slope). The equation of this line is y = 3 since y is 3 everywhere along the line.

To find the average rate of change of a function between any two points on its graph, we calculate the slope of the line containing the two points. (7, 8) (0, 0) So the average rate of change of this function between the points (0, 0) and (7, 8) is 8/7 Slope of this line

S C A T T E R D I A G R A M S LINEAR CURVE FITTING A scatter diagram is a plot of ordered pairs generally obtained by observation of some relation.

Look at these scatter diagrams to see if the relation looks linear or nonlinear. LINEAR NONLINEAR LINEAR NONLINEAR LINEARNONLINEAR

Let's look at some data gathered about the relationship between the speed of a certain car and the miles per gallon it gets speed x MPG y ordered pair (x, y) We can plot these points and see if it looks like there is a relationship It looks like a linear relationship because as you look it seems to have the pattern of a line with negative slope

We could come up with a function to estimate the miles per gallon given the speed. We'll pick two points on or near the line we made and find the slope and then use the point-slope formula. We'll choose (52, 28) and (65, 17) (52, 28) (65, 17) Find the estimated MPG if the car speed is 70 mph mpg

Click here Click here for help on how to use your calculator to create a scatter diagram and find the line of best fit. We could each come up with a slightly different line if we picked two different points to use. There is a process for finding the best line. This process is covered in a statistics class. We'll just use the calculator to find what is called the "line of best fit".

Properties of the Correlation Coefficient, r ≤ 1 2. When r > 0, there is a positive linear correlation 3. When r < 0, there is a negative linear correlation 4. When | r | ≈ 0, there is weak or no linear correlation.

If a > 0 the parabola opens up and the larger the a value the “narrower” the graph and the smaller the a value the “wider” the graph. If a < 0 the parabola opens down and the larger the  a  the “narrower” the graph and the smaller the  a  the “wider” the graph.

vertical shift, moves graph vertically by k horizontal shift, moves graph horizontally by h Determines whether the parabola opens up or down and how “wide” it is We need to algebraically manipulate this to look like the form above. We’ll do this by completing the square. Add a number here to make a perfect square Subtract it here to keep things equal (can’t add a number without compensating for it and we don’t want to add it to the other side because of function notation) 99 This will factor into (x-3)(x-3) so we can express it as something squared and combine the -1 and -9 on the end. The graph of this function is a parabola Let’s look at a quadratic function and see if we can graph it.

right 3 down 10 We started with and completed the square to get it in the format to be able to graph using transformations. We can take the general quadratic equation and do this to find a formula for the vertex. (This is done in your book). What we find from doing this is on the next slide.

The x value of the vertex of the parabola can be found by computing The y value of the vertex of the parabola can be found by substituting the x value of the vertex in the function and finding the function value. Let’s try this on the one we did before: 1 (1) (-6) The vertex is then at (3, -10)

(3, -10) Let’s plot the vertex: Since the a value is positive, we know the parabola opens up. The parabola will be symmetric about a vertical line through the vertex called the axis of symmetry. Let’s find the y intercept by plugging 0 in for x. So y intercept is (0, -1) The graph is symmetric with respect to the line x = 3 so we can find a reflective point on the other side of the axis of symmetry. (0, -1)(6, -1) We can now see enough to graph the parabola

(3, -10) Let’s look at another way to graph the parabola starting with the vertex: We could find the x intercepts of the graph by putting f(x) (which is the y value) = 0 This won’t factor so we’ll have to use the quadratic formula. So x intercepts are (6.2, 0) and (- 0.2, 0)

A mathematical model may lead to a quadratic function. Often, we are interested in where the function is at its minimum or its maximum. If the function is quadratic the graph will be a parabola so the minimum (if it opens up) will be at the vertex or the maximum (if it opens down) will be at the vertex. We can find the x value of the vertex by computing We could then sub this value into the function to find its minimum or maximum value.

DEMAND EQUATION The price p and the quantity x sold of a certain product obey the demand equation: This is the real world domain. The equation doesn’t make sense if the quantity sold is negative (x 400) Express the revenue R as a function of x. Revenue is the amount you bring in, so it would be how much you charge (the price p) times how many you sold (the quantity x)

This is a quadratic equation and since a is negative, its graph is a parabola that opens down. It will have a maximum value then at the y value of the vertex. What is the revenue if 100 units are sold? What quantity x maximizes revenue? Since the revenue function is maximum at the vertex, we'll want to find the x value of the vertex to answer this. What is the maximum revenue? This would be the y value of the vertex

DEMAND EQUATION The price p and the quantity x sold of a certain product obey the demand equation: What price should the company charge to receive maximum revenue? Since we just found that the quantity to achieve maximum revenue was 200, we can substitute this in the price equation to answer this question.