Problem 36 - Algebra for College Students

Question

For Problems \(35-42\), (a) find the \(y\) intercepts, (b) find the \(x\) intercepts, and (c) find the intervals of \(x\) where \(f(x)>0\) and those where \(f(x)<0\). Do not sketch the graphs. $$ f(x)=(x-5)(x+4)(x-3) $$

Step-by-Step Solution

Answer

(a) y-intercept: (0,60); (b) x-intercepts: (5,0), (-4,0), (3,0); (c) f(x)>0: (-4,3), (5,∞), f(x)<0: (-∞,-4), (3,5).

Step 1: Identify the y-intercept

The y-intercept occurs where the function intersects the y-axis, which is when \(x=0\). Substitute \(x=0\) into \(f(x)\) to find the y-intercept: \ \( f(0) = (0-5)(0+4)(0-3) = (-5)(4)(-3) = 60 \). \ Thus, the y-intercept is \((0, 60)\).

Step 2: Find the x-intercepts

The x-intercepts occur where the function intersects the x-axis, which is when \(f(x)=0\). Set the function \((x-5)(x+4)(x-3)\) equal to zero and solve for \(x\): \ - \(x-5=0 \Rightarrow x=5\) \ - \(x+4=0 \Rightarrow x=-4\) \ - \(x-3=0 \Rightarrow x=3\) \ Thus, the x-intercepts are \((5,0)\), \((-4,0)\), and \((3,0)\).

Step 3: Determine intervals where \(f(x)>0\) or \(f(x)<0\)

To find where \(f(x)>0\) or \(f(x)<0\), analyze the sign changes across the x-intercepts \(-4\), \(3\), and \(5\). Evaluate the function over the intervals: \ - For \(x<-4\), choose \(x=-5\): \((x-5)(x+4)(x-3) < 0\) \ - For \(-4 0\) \ - For \(35\), choose \(x=6\): \((x-5)(x+4)(x-3) > 0\) \ Thus, \(f(x) > 0\) in the intervals \((-4,3)\) and \((5, \infty)\), and \(f(x) < 0\) in the intervals \(( -\infty, -4)\) and \((3,5)\).

Key Concepts

y-intercept

A y-intercept is the point where a function crosses the y-axis. This happens when the x-value is zero. To find the y-intercept of a polynomial function, you simply set the variable \( x \) to zero in the function and solve for \( f(x) \). It is an important feature because it tells you where the graph intersects the y-axis, giving a starting clue about the function's behavior.
For instance, considering the function \( f(x) = (x-5)(x+4)(x-3) \), substituting \( x=0 \) yields \( f(0) = (-5)(4)(-3) = 60 \).
This computation shows that the y-intercept is \((0, 60)\). Here, \( 60 \) represents the function's value when \( x \) is zero.

x-intercept

The x-intercepts of a function are its solutions when the output is zero, or when \( f(x) = 0 \). In a graph, these are the points where the function crosses the x-axis. To find them, solve the equation for when the whole expression equals zero. For polynomial functions, this means identifying the values of \( x \) that make each factor of the polynomial equal to zero.
For example, in \( f(x)=(x-5)(x+4)(x-3) \):

• Set \( x-5 = 0 \), which gives \( x = 5 \)
• Set \( x+4 = 0 \), which gives \( x = -4 \)
• Set \( x-3 = 0 \), which gives \( x = 3 \)

Thus, the x-intercepts are \((5,0)\), \((-4,0)\), and \((3,0)\). These points indicate where the graph touches or crosses the x-axis.

polynomial functions

Polynomial functions are expressions that consist of variables raised to whole number powers, combined with coefficients. They are versatile and can model a wide array of real-world scenarios. A basic polynomial function can look like \( f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \).
The degree of a polynomial is the highest power of \( x \) present. For example, in \( f(x) = (x-5)(x+4)(x-3) \), if expanded, the highest power of \( x \) determines its degree.
Polynomials have unique characteristics based on their degree. For instance:

• They can have multiple intercepts.
• The degree informs potential changes in direction.
• The leading term primarily determines the end behavior.

Recognizing the structure of a polynomial helps in predicting its graph and behavior across different values of \( x \).

intervals of positivity and negativity

Identifying intervals where a function is positive or negative provides insight into the behavior of polynomial functions between its intercepts. This process involves examining the sign of the function over different x-intervals, divided by its intercepts.
To find where \( f(x) > 0 \) or \( f(x) < 0 \), consider the areas around each x-intercept. For \( f(x)=(x-5)(x+4)(x-3) \), divide the x-axis into intervals based on intercepts \(-4\), \(3\), and \(5\).
Test points within each interval:

• \( x < -4 \): Choose \( x = -5 \). The function is negative.
• \(-4 < x < 3 \): Choose \( x = 0 \). The function is positive.
• \( 3 < x < 5 \): Choose \( x = 4 \). The function is negative.
• \( x > 5 \): Choose \( x = 6 \). The function is positive.

Thus, the function is positive in \((-4, 3)\) and \((5, \infty)\), and negative in \((-\infty, -4)\) and \((3, 5)\).
This step helps you sketch or understand the function's overall shape without graphing.