How do you graph #f(x)=x^4-4# using zeros and end behavior?

Question

Sadie Ackerman · Accepted Answer

Find the zeros, end behavior and y intercept as described below.

Graph #f(x)=x^4-4# using zeros and end behavior.

To find the zeros, factor the polynomial.

#f(x)=(x^2-2)(x^2+2)#

Factor again.

#f(x)=(x+sqrt2)^color(red)1(x-sqrt2)^color(red)1(x^2+2)#

Setting each factor equal to zero and solving gives:

#x=-sqrt2#, #x=sqrt2# and #x=+-sqrt2i#

The only real zeros are #sqrt2# and #-sqrt2#. Each has a multiplicity of #color(red)1# because the exponent on each factor is #color(red)1#. An odd multiplicity means the graph crosses (or cuts through) the #x# axis at the zeros/x-intercepts. The #x# intercepts are #(-sqrt2,0)# and #(sqrt2, 0)# which are approximately #(+-1.414,0)#.

To find the end behavior, examine the degree and leading coefficient of the original polynomial.

#f(x)=color(blue)1x^color(violet)4-4#

The degree is #color(violet)4# and the leading coefficient is #color(blue)1#.

An even degree with a positive leading coefficient indicates that as#xrarroo# and #xrarr-oo#, #f(x)rarroo#. In other words, the "ends" of the graph both point "up".

It is also helpful to find the #y# intercept. Setting #x=0# gives
#y=0^4-4=-4#. The #y# intercept is #(0, -4)#

The graph is shown below.

Lily Anderson · Answer

undefined To graph the function f(x) = x^4 - 4 using zeros and end behavior, follow these steps:

1. Find the zeros of the function by setting f(x) equal to zero and solving for x.
   x^4 - 4 = 0
   x^4 = 4
   x = ±√2 and x = ±√(-2) (since x^4 = (-x)^4)
   The real zeros are x = √2 and x = -√2.

2. Determine the end behavior of the function as x approaches positive and negative infinity. Since the leading term of the polynomial is x^4, the end behavior is the same as that of a fourth-degree polynomial, which means:
   - As x approaches positive infinity, f(x) approaches positive infinity.
   - As x approaches negative infinity, f(x) approaches positive infinity.

3. Plot the zeros (√2, 0) and (-√2, 0) on the graph.

4. Sketch the graph of the function approaching positive and negative infinity based on the end behavior determined in step 2.

Combining these steps, you can sketch the graph of f(x) = x^4 - 4 using the zeros (√2, 0) and (-√2, 0) and considering its end behavior.