In this example we're given the factored form of a polynomial function and asked to find the y intercept, the x intercepts, which could also be referred to as the real zeros or real roots of the polynomial function, and then also asked to describe the end behavior of the polynomial function.
We should recognize the given polynomial function as a degree 2 polynomial function, and therefore, we can also call this a quadratic function. If it's helpful, we can replace f of x with y and write this as y = the quantity x + 2 x the quantity x 4.
And now to find the intercepts of any function, the process is the same. To find the y intercept, we set x = to 0 and solve for y.
And to find the x intercept, so the real zeros, we set y = to 0 and solve for x. Let's go ahead and do this on the next slide. So, again, to find the y intercept, set x = to 0 and solve for y.
And let's go ahead and use this form of the equation. So we would have y = the quantity 0 + 2 x the quantity 0 4. That would give us y = +2 x -4, which is = to -8.
So the y intercept = -8. And now to find the x intercepts, we'll set y = to 0 and solve for x. So that would give us the equation 0 = the quantity x + 2 x the quantity x 4.
Well, this product here on the right would be = to 0 when x + 2 = 0, or when x 4 = 0. Solving for x here, we'd subtract 2 on both sides, so = -2.
Adding 4 on both sides we'd have x = +4. So the x intercepts = -2 and +4. So going back to the previous slide, the y intercept is -8. We could give this as an ordered pair where the x coordinate would be 0 and the y coordinate is -8.
And we add 2 x intercepts. We had x = -2 and x = +4 that's ordered pairs. This would be two points (-2, 0) and the point (4, 0). And now to describe the end behavior we want to describe the value of the function or the y value as x approaches the left or negative infinity, and as x approaches right or positive infinity.
To verify the end behavior we will look at the graph of the function in just a moment, but we should also recognize the leading term of this polynomial function would be x squared.
So if we have a function in the form of f of x = x squared, and then plus or minus other terms, we should be able to describe the end behavior just by using the first term of the polynomial function.
We should recognize the polynomial function is a leading term of 1x squared would be a parabola that opens upward like this. And therefore, as x approaches to the right or positive infinity, notice how the y values approach positive infinity to the right and to the left.
But, again, to verify this, let's go ahead and graph the given function. Let's go ahead and press y equals, clear out any old functions, and type in our new function. So we have the quantity x + 2 x the quantity x 4.
We'll go ahead and press zoom 6 for the standard window. And now as x approaches positive infinity we'd be moving right along the graph. Notice as we move right approaching positive infinity the function values increase without bound or go upward.
And therefore, the function values or the y values approach positive infinity, and the same is true as we approach the left or as x approaches negative infinity. As we move to the left, we can see the function moving upward without bound, and therefore the y values or function values are approaching positive infinity.
So as x approaches infinity f of x approaches infinity or positive infinity. And as x approaches negative infinity f of x or y also approaches positive infinity.
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