Problem 265 - Precalculus

Question

For the following exercises, graph the potynomial functions. Note \(x\) - and \(y\) -intercepts, multiplicity, and end behavior. $$ g(x)=(x+4)(x-1)^{2} $$

Step-by-Step Solution

Answer

x-intercepts: -4 (crosses), 1 (touches); y-intercept: (0, 4); End behavior: \(\pm\infty\).

Step 1: Find the x-intercepts

To find the x-intercepts, set the function equal to zero and solve for \(x\): \ \(g(x) = (x + 4)(x - 1)^2 = 0\). \ The x-intercepts occur when each factor equals zero: \ \(x + 4 = 0 \Rightarrow x = -4\) and \ \((x - 1)^2 = 0 \Rightarrow x = 1\). \ So, the x-intercepts are \(x = -4\) and \(x = 1\).

Step 2: Determine the multiplicity of the x-intercepts

The multiplicity of an x-intercept is determined by the exponent on its corresponding factor. For \(x = -4\), the factor is \((x + 4)^1\), so its multiplicity is 1 (odd). For \(x = 1\), the factor is \((x - 1)^2\), so its multiplicity is 2 (even).

Step 3: Find the y-intercept

The y-intercept occurs where \(x = 0\). Substitute \(x = 0\) into the function: \ \(g(0) = (0 + 4)(0 - 1)^2 = 4 \times 1 = 4\). \ Therefore, the y-intercept is \( (0, 4) \).

Step 4: Determine the end behavior

The end behavior of a polynomial is determined by its leading term. Expand the expression \(g(x) = (x + 4)(x - 1)^2\): \ \((x + 4)(x^2 - 2x + 1) = x^3 + 2x^2 - 7x + 4\). \ The leading term is \(x^3\), which indicates the end behavior. \ As \(x \rightarrow \infty\), \(g(x) \rightarrow \infty\) and as \(x \rightarrow -\infty\), \(g(x) \rightarrow -\infty\).

Step 5: Sketch the graph

Plot the x-intercepts \((-4, 0)\) and \((1, 0)\), the y-intercept \((0, 4)\), and outline the end behavior. Remember the x-intercept at \(-4\) crosses the x-axis due to odd multiplicity and the intercept at \(1\) touches the x-axis but does not cross it due to even multiplicity. The leading term \(x^3\) guides the ends going in opposite directions.

Key Concepts

x-intercepts

When graphing polynomial functions, finding the x-intercepts is essential. These are the points on the graph where the function crosses the x-axis. In mathematical terms, this is where the value of the function is zero. For the polynomial function \(g(x) = (x + 4)(x - 1)^2\), you set the function equal to zero and solve for \(x\), giving us:

• \(x + 4 = 0\), thus \(x = -4\)
• \((x - 1)^2 = 0\), thus \(x = 1\)

This means the function has two x-intercepts: \(x = -4\) and \(x = 1\). By identifying these x-intercepts, you can better understand where the graph intersects the x-axis and begin outlining the shape of the graph.

y-intercepts

The y-intercept of a graph is where the graph intersects the y-axis. At this point, the value of \(x\) is zero. To find the y-intercept of the function \(g(x)=(x+4)(x-1)^2\), plug in \(x = 0\) into the function:\[g(0) = (0 + 4)(0 - 1)^2 = 4 \times 1 = 4\]Hence, the y-intercept of the graph is at the point \((0, 4)\). This tells you where the graph will touch or cross the y-axis, which is vital for creating accurate plots when graphing polynomial functions.

multiplicity

Multiplicity refers to how many times a particular x-intercept appears as a root of the function. It influences how the graph behaves at the intercepts:

• A factor with an odd multiplicity, like \((x+4)^1\), means the graph will cross the x-axis at that point.
• A factor with an even multiplicity, like \((x-1)^2\), means the graph touches but doesn’t cross the x-axis.

In our example, the x-intercept at \(x = -4\) has a multiplicity of 1, which is odd, so the graph crosses the x-axis there. The x-intercept at \(x = 1\) has a multiplicity of 2, which is even, indicating the graph touches and bounces off the x-axis at this point. Understanding multiplicity helps in predicting the shape and direction of the graph around these intercepts.

end behavior

End behavior describes what happens to the value of a polynomial function as \(x\) approaches positive or negative infinity. The end behavior is primarily determined by the leading term of the polynomial. When we expand \(g(x) = (x+4)(x-1)^2\) to \(x^3 + 2x^2 - 7x + 4\), we see the leading term is \(x^3\).With a cubic leading term:

• As \(x \rightarrow \infty\), \(g(x) \rightarrow \infty\).
• As \(x \rightarrow -\infty\), \(g(x) \rightarrow -\infty\).

This pattern of end behavior tells us that on the right side, the graph shoots up as \(x\) becomes large and positive. On the left side, the graph goes downward as \(x\) becomes large and negative. Understanding end behavior is crucial for predicting how the graph extends beyond the intercepts.