Match each of the graphs to the polynomial equation it represents. For the graph without a matching equation, write down what must be true about the polynomial equation.
A
Blank coordinate plane, X axis and Y axis, origin covered with a dashed square. Polynomial graph comes from Quadrant 2 upper left, descends to the square box, comes out in Quadrant 1 and ascends to upper right.
B
Blank coordinate plane, X axis and Y axis, origin covered with a dashed square. Polynomial graph comes from Quadrant 3, horizontally to the square box and comes out in Quadrant 4 and horizontally to the right.
C
Blank coordinate plane, X axis and Y axis, origin covered with a dashed square. Polynomial graph comes from Quadrant 2 upper left, descending to the square box and comes out in Quadrant 4 and descends to lower right.
D
Blank coordinate plane, X axis and Y axis, origin covered with a dashed square. Polynomial graph comes from Quadrant 3 lower left, ascending to the square box and comes out in Quadrant 1 and ascends to upper right.
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
For , which function do you think has greater values? Explain your reasoning.
Student Lesson Summary
Consider the polynomial functions and . For any non-zero real number , the output of is positive while the output of is negative. The signs of all the output values for are the opposite of those of , since the product of any number and a negative number has a sign opposite of the original number. The difference between these two functions is also easy to see when we look at their graphs.
Graph of polynomial function y = -x squared, xy-plane, origin O. Horizontal axis, scale -75 to 75, by 25’s. Vertical axis, scale -4,000 to 4,000, by 1,000’s. Polynomial graph comes from Quadrant 3, passes through about (-63 point 3 comma -4,000), ascends in a smooth curve to the origin, into Quadrant 4 and descends in a smooth curve to the lower right through about (63 point 3 comma -4,000).
Graph of polynomial function y = x squared, xy-plane, origin O. Horizontal axis, scale -75 to 75, by 25’s. Vertical axis, scale -4,000 to 4,000, by 1,000’s. Polynomial graph comes from Quadrant 2, passes through about (-63 point 3 comma 4,000), descends in a smooth curve to the origin, into Quadrant 4 and ascends in a smooth curve to the upper right through about (63 point 3 comma 4,000).
Now consider the graphs of , which has a leading term of , and , which has a leading term of . They have the same zeros, but opposite end behavior, because they have opposite signs on their leading coefficients.
Graph of a polynomial, xy-plane, origin O. Horizontal axis, scale -75 to 75, by 25’s. Vertical axis, scale -4,000 to 4,000, by 1,000’s. Polynomial graph comes from Quadrant 2 passes through about (-14 point 3 comma 4,000) descends to the origin, into Quadrant 4 and descends to lower right through about (11 point 3 comma -4,000).
Graph of a polynomial, xy-plane, origin O. Horizontal axis, scale -75 to 75, by 25’s. Vertical axis, scale -4,000 to 4,000, by 1,000’s. Polynomial graph comes from Quadrant 3 passes through about (-14 point 3 comma -4,000) ascends to the origin, into Quadrant 1 and ascends to upper right through about (11 point 3 comma 4,000).
For polynomials of both even and odd degree, a negative leading coefficient results in the end behavior of the polynomial being the opposite of what it would be if the leading coefficient were positive.
Glossary
None
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