Problem 8 Graph the function. $$f(x)=(x+... [FREE SOLUTION]

Chapter 4: Problem 8

Graph the function. $$f(x)=(x+2)^2(x+4)^2$$

Short Answer

Expert verified

The y-intercept of the function is at (0, 64); the zeros are at -2 and -4; the end behavior indicates the function rising to the left and to the right; and finally, the graph of the function is a curve starting from the second quadrant, touching the x-axis at points (-4,0) and (-2,0), and then shooting back up into the first quadrant completing the graph.

Step by step solution

Identifying the y-intercept

The y-intercept of a function refers to the point where the curve intersects the y-axis, i.e., when x = 0. So, substitute $ x = 0 $ in the function to get $ f(0) = (0+2)^2(0+4)^2 = 4*16 = 64 $. Thus, the y-intercept is at (0, 64).

Finding the zeros of the function

The zeros or roots of the function are the values at which the function value $ f(x) $ is 0. To find the zeros of the given function, equate the function $ f(x) = (x+2)^2(x+4)^2 $ to 0 and solve for x. This results in $ (x+2)^2 = 0 $ and $ (x+4)^2 = 0 $. The solutions are $ x = -2, -4 $. Thus, the function has zeros at $ x = -2 $ and $ x = -4 $.

Identifying the end behavior

The end behavior of a function refers to what happens to the function values $ f(x) $ as $ x $ approaches positive or negative infinity. Since the highest degree of the function is 4, a positive even number, and the leading coefficient (coefficient of highest degree term) is positive, the function will rise to the right and to the left.

Sketching the graph

Start by plotting the y-intercept at the point (0, 64). Next, plot the zeros of the function at $ x = -2, -4 $. These points divide the x-axis into segments. Since the function is a 4th degree with positive leading coefficient, the graph will start from the top quadrant, come down to touch or cross the x-axis at $ x = -4, -2 $ and then go back up. Over these segments, the function is either above or below the x-axis, changes sign at $ x = -4 $ and $ x = -2 $ because these are double zeros (or roots of even multiplicity) causing the graph to touch the x-axis and then turn back again. Make sure the graph reflects the end behaviors.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Y-Intercept of a Function

When graphing functions, pinpointing the y-intercept is a critical initial step. The y-intercept is the point where the curve meets the y-axis, which corresponds to the output value of the function when the input, or x-value, is zero. To find this, you simply substitute zero for all the occurrences of x in the function and calculate the result. For the polynomial function given as an example, $ f(x)=(x+2)^2(x+4)^2 $, substituting zero gives us $ f(0)=4^2 \times 4^2=16 \times 16=256 $. It's important to note that the y-intercept provides a valuable reference point for graphing because it gives us an assured point on the curve.

Zeros of a Function

Another crucial aspect of graphing is determining the zeros, or roots, of the function. These are the points where the function's value is zero, indicating where the curve will intersect the x-axis. To find these points, one sets the function equal to zero and solves for x. For instance, the zeros of the function $ f(x)=(x+2)^2(x+4)^2 $ are found by solving the equation $ (x+2)^2=0 $ and $ (x+4)^2=0 $ thus, revealing the zeros at $ x = -2 $ and $ x = -4 $. The zeros provide a foundational framework for the graph, segmenting the x-axis and informing us about the intervals in which the function exists above or below the x-axis.

End Behavior of Polynomial

The end behavior of a polynomial function refers to the direction in which the graph of the function extends as x approaches positive or negative infinity. This characteristic offers a big picture overview of how the function behaves outside the limited view of a graphing window. For the given function, $ f(x) $ is a fourth-degree polynomial with a positive leading coefficient, which means the graph will rise as x approaches infinity and also as x approaches negative infinity. This behavior is because polynomials with even degrees and positive leading coefficients will always rise to the top in both directions, resembling the edges of a 'U' shape.

Polynomial Root Multiplicity

Root multiplicity is the number of times a particular root appears in a polynomial equation. It has a profound impact on the graph of the function at that root鈥檚 intercept. For our example, $ f(x) $ has zeros at $ x = -2 $ and $ x = -4 $, both of which are of multiplicity two, meaning they are repeated roots. On the graph, this results in the curve 鈥榖ouncing鈥� off the x-axis at these points rather than crossing it. When a polynomial has a root with even multiplicity, the graph will touch the axis at this point, but it will not pass through; it typically signifies a local minima or maxima at these points, depending on the function's orientation.

91影视

Graph the function. $$f(x)=(x+2)^2(x+4)^2$$

Short Answer

Step by step solution

Identifying the y-intercept

Finding the zeros of the function

Identifying the end behavior

Sketching the graph

Key Concepts

Y-Intercept of a Function

Zeros of a Function

End Behavior of Polynomial

Polynomial Root Multiplicity

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