Describe the end behavior of the graph of the polynomial function. Then evaluate the function for x=-4, -3, -2, -1, 0, 1, 2, 3, 4. Then graph the function.y=x^4-2x^2-x-1 and y=(x-3)(x+1)(x+2)
Did you try just following the instructions?
They seem pretty straightforward.
For the end results, try ± large values
e.g. let x=1000 and x = -1000
wouldn't both x^4 become huge for both values, so what happens to the y value at those far-out values of x ?
notice that the second function starts with y = x^3 ...
when x = 1000 , y = +1000000000
when x = -1000 , y = (-1000)^3 = -1000000000
what does that tell you about the position of y for huge negative values of x ?
p(x)=x^3+6x^2+5x-12 Describe the end behavior of the function
To determine the end behavior of the graph of the polynomial function, you need to look at the degree of the polynomial. In both cases, the degree of the polynomial is 4.
For the function y = x^4 - 2x^2 - x - 1, the leading coefficient (the coefficient of the term with the highest degree) is positive (+1). This means that as x approaches negative infinity (-∞), the function value approaches positive infinity (+∞), and as x approaches positive infinity (+∞), the function value also approaches positive infinity (+∞).
Now, let's evaluate the function y = x^4 - 2x^2 - x - 1 for the given x-values:
For x = -4:
y = (-4)^4 - 2(-4)^2 - (-4) - 1
y = 256 - 2(16) + 4 - 1
y = 256 - 32 + 4 - 1
y = 227
For x = -3:
y = (-3)^4 - 2(-3)^2 - (-3) - 1
y = 81 - 2(9) + 3 - 1
y = 81 - 18 + 2
y = 65
For x = -2:
y = (-2)^4 - 2(-2)^2 - (-2) - 1
y = 16 - 2(4) + 2 - 1
y = 16 - 8 + 1
y = 9
For x = -1:
y = (-1)^4 - 2(-1)^2 - (-1) - 1
y = 1 - 2(1) + 1 - 1
y = 1 - 2 + 1 - 1
y = -1
For x = 0:
y = (0)^4 - 2(0)^2 - (0) - 1
y = 0 - 0 - 0 - 1
y = -1
For x = 1:
y = (1)^4 - 2(1)^2 - (1) - 1
y = 1 - 2 + 1 - 1
y = -1
For x = 2:
y = (2)^4 - 2(2)^2 - (2) - 1
y = 16 - 2(4) - 2 - 1
y = 16 - 8 - 2 - 1
y = 5
For x = 3:
y = (3)^4 - 2(3)^2 - (3) - 1
y = 81 - 2(9) - 3 - 1
y = 81 - 18 - 3 - 1
y = 59
For x = 4:
y = (4)^4 - 2(4)^2 - (4) - 1
y = 256 - 2(16) - 4 - 1
y = 256 - 32 - 4 - 1
y = 219
Finally, to graph the function y = x^4 - 2x^2 - x - 1, plot the points you evaluated and connect them smoothly to create a graph.
Now, for the function y = (x-3)(x+1)(x+2), we can find the x-intercepts which are the values that make the function equal to zero:
Setting y = 0:
(x-3)(x+1)(x+2) = 0
Solving for x:
x-3 = 0 or x+1 = 0 or x+2 = 0
x = 3 or x = -1 or x = -2
So the x-intercepts are x = 3, x = -1, and x = -2.
To graph the function y = (x-3)(x+1)(x+2), plot the x-intercepts at (3, 0), (-1, 0), and (-2, 0). Since the degree of this function is odd, the end behavior is different from the previous one. As x approaches negative infinity (-∞), the function value also approaches negative infinity (-∞), and as x approaches positive infinity (+∞), the function value approaches positive infinity (+∞).
Connect the points to create the graph of y = (x-3)(x+1)(x+2).