Zeros And End Behavior Of F(x)=(x-1)(x+2)(x-3)

Understanding the Zeros of a Function

Let's dive into finding the zeros of the function $f(x)=(x-1)(x+2)(x-3)$. The zeros of a function, also known as roots, are the values of $x$ for which the function equals zero, i.e., $f(x) = 0$. In simpler terms, these are the points where the graph of the function crosses or touches the x-axis. For a polynomial function expressed in factored form, like our $f(x)$, finding the zeros is quite straightforward. We simply set each factor equal to zero and solve for $x$. So, for our function, we have three factors: $(x-1)$, $(x+2)$, and $(x-3)$.

Setting the first factor to zero, we get $x-1=0$, which gives us $x=1$. This means that $x=1$ is one of the zeros of the function. When $x$ is $1$, the function's output $f(x)$ is $0$. Now, let's look at the second factor, $(x+2)$. Setting this to zero, we have $x+2=0$, and solving for $x$ yields $x=-2$. Therefore, $x=-2$ is another zero of our function. Finally, we consider the third factor, $(x-3)$. Setting $x-3=0$, we find that $x=3$ is the last zero of the function. So, the zeros of the function $f(x)=(x-1)(x+2)(x-3)$ are $1$, $-2$, and $3$. It's important to remember that each of these values makes the entire function's output zero because any number multiplied by zero results in zero.

Exploring the End Behavior of Polynomials

Now, let's shift our focus to the end behavior of the function $f(x)=(x-1)(x+2)(x-3)$. The end behavior of a polynomial describes what happens to the function's output (y-values) as the input (x-values) approach positive or negative infinity. For polynomial functions, the end behavior is primarily determined by the degree of the polynomial and the sign of its leading coefficient. The degree of a polynomial is the highest power of $x$ when the polynomial is expanded, and the leading coefficient is the coefficient of that term.

In our case, $f(x)=(x-1)(x+2)(x-3)$ is a polynomial. If we were to expand it, we would find the highest power of $x$ comes from multiplying the $x$ terms from each factor: $x * x * x = x^3$. This means our polynomial has a degree of $3$. Since the degree is odd, the end behavior will be opposite on either side: one end will go up, and the other will go down. Now, let's determine the sign of the leading coefficient. Again, by expanding the factored form, the term with the highest power of $x$ will be $x^3$. The coefficient of this $x^3$ term is positive ($+1$). A positive leading coefficient with an odd degree tells us a specific pattern for the end behavior. As $x$ approaches positive infinity (moves far to the right on the graph), the function's values $f(x)$ will also approach positive infinity (go upwards). Conversely, as $x$ approaches negative infinity (moves far to the left on the graph), the function's values $f(x)$ will approach negative infinity (go downwards).

This means that the graph of $f(x)=(x-1)(x+2)(x-3)$ starts from the bottom left and goes up towards the top right. You can visualize this by imagining plugging in very large positive and very large negative numbers for $x$. For a large positive $x$, all the factors $(x-1)$, $(x+2)$, and $(x-3)$ will be positive, resulting in a large positive $f(x)$. For a large negative $x$, $(x-1)$ will be negative, $(x+2)$ will be negative, and $(x-3)$ will be negative. The product of three negative numbers is negative, so $f(x)$ will be a large negative number. This confirms our prediction: downward to the left and upward to the right.

Connecting Zeros and End Behavior

Understanding both the zeros and the end behavior of a function provides a fundamental framework for sketching its graph. We've identified the zeros of $f(x)=(x-1)(x+2)(x-3)$ as $1$, $-2$, and $3$. These are the specific points where the graph will intersect the x-axis. The end behavior tells us the overall direction of the graph as we move away from the origin. We determined that the function starts low on the left (as $x$ approaches $-\infty$, $f(x)$ approaches $-\infty$) and ends high on the right (as $x$ approaches $+\infty$, $f(x)$ approaches $+\infty$).

Let's piece this information together. The graph must pass through the x-axis at $x=-2$, $x=1$, and $x=3$. Since the function starts by coming from the bottom left, it must cross the x-axis at $x=-2$ to become positive. After crossing at $x=-2$, the function must eventually turn around (this is where calculus helps to find local extrema, but we don't need it for end behavior and zeros alone) and come back down to cross the x-axis again at $x=1$. After crossing at $x=1$, the function must again turn around and head upwards, crossing the x-axis a third time at $x=3$. Once it crosses $x=3$, it continues to increase towards positive infinity, aligning perfectly with our predicted end behavior of going upward to the right. This sequence of crossings and the overall directional trend gives us a solid understanding of the function's shape and trajectory.

The number of zeros a polynomial has (counting multiplicity) is at most equal to its degree. Our function has a degree of $3$, and we found $3$ distinct real zeros. This is the maximum possible for a degree $3$ polynomial. The factored form directly reveals these zeros, making it a powerful representation for analysis. The end behavior, dictated by the leading term ($x^3$ in this case), provides the essential boundary conditions for the graph. Without understanding these two aspects, sketching a polynomial graph would be a much more complex task, involving finding critical points and intervals of increase and decrease. However, with the zeros and end behavior, we can confidently sketch a general shape of the curve that accurately represents the function's key characteristics.

Consider the behavior between the zeros. For instance, between $x=-2$ and $x=1$, the function must be positive. We can test a value, say $x=0$: $f(0) = (-1)(2)(-3) = 6$, which is indeed positive. Between $x=1$ and $x=3$, the function must be negative. Let's test $x=2$: $f(2) = (2-1)(2+2)(2-3) = (1)(4)(-1) = -4$, which is negative. These checks further validate our understanding derived from the zeros and end behavior. This comprehensive approach ensures that our interpretation of the function's graphical representation is accurate and complete, forming a solid foundation for further mathematical exploration. The interplay between where the function crosses the x-axis and its ultimate trajectory as x tends towards infinity is a fundamental concept in understanding polynomial functions.

Final Answer Summary

To summarize, for the function $f(x)=(x-1)(x+2)(x-3)$:

• Zeros: The values of $x$ for which $f(x)=0$ are found by setting each factor to zero. This gives us $x=1$, $x=-2$, and $x=3$. These are the points where the graph intersects the x-axis.
• End Behavior: The function is a polynomial of degree $3$ with a positive leading coefficient. As $x$ approaches negative infinity, $f(x)$ approaches negative infinity (continues downward to the left). As $x$ approaches positive infinity, $f(x)$ approaches positive infinity (continues upward to the right). This means the graph starts from the bottom left and moves towards the top right.

Therefore, the correct option that describes the zeros and end behavior is one that lists the zeros as $1, -2, 3$ and states that the function continues downward to the left and upward to the right.

For further exploration into the fascinating world of polynomial functions, you can visit resources like Khan Academy's section on polynomial functions. It offers a wealth of information, examples, and practice exercises that can deepen your understanding of these essential mathematical concepts.