Question

For each function, determine i) the \(x\) -intercepts of the graph ii) the degree and end behaviour of the graph iii) the zeros and their multiplicity iv) the \(y\) -intercept of the graph v) the intervals where the function is positive and the intervals where it is negative a) \(y=x^{3}-4 x^{2}-45 x\) b) \(f(x)=x^{4}-81 x^{2}\) c) \(h(x)=x^{3}+3 x^{2}-x-3\) d) \(k(x)=-x^{4}-2 x^{3}+7 x^{2}+8 x-12\)

Short answer

The x-intercepts are the roots of the equation. The degree determines end behavior. Zeros and their multiplicities are found by factoring. The y-intercept is found by setting x=0. Positive and negative intervals are determined with a sign chart.

Step-by-step solution

- Finding x-Intercepts

For each function, set the equation equal to zero and solve for x to find the x-intercepts. For example, for the function y=x^3-4x^2-45x, set x^3 - 4x^2 - 45x = 0 and solve for x.

- Determining the Degree and End Behavior

The degree of a polynomial is the highest power of x in its expression. To determine the end behavior, observe the leading term (the term with the highest power). For example, in y=x^3 - 4x^2 - 45x, the degree is 3. Since the coefficient of the leading term is positive, as x -> ∞, y -> ∞ and as x -> -∞, y -> -∞.

- Finding Zeros and Their Multiplicities

The zeros are the x-values where the function equals zero. Factorize the function to find the zeros and their multiplicities. For multiplicities, observe the power of the corresponding factor. For example, y=x^3-4x^2-45x can be factorized as y=x(x-9)(x+5); hence the zeros are x=0, x=9, and x=-5, each with multiplicity 1.

- Finding y-Intercept

To find the y-intercept, substitute x=0 in the function and solve for y. For example, for y=x^3-4x^2-45x, substituting x=0 gives y(0)=0.

- Determine Positive and Negative Intervals

To find intervals where the function is positive or negative, use a sign chart. Select test points from the intervals divided by the zeros and substitute these points into the function to check whether the function is positive or negative. For example, for y=x^3-4x^2-45x with zeros at x=-5, 0, and 9; test points might be in intervals (-∞, -5), (-5, 0), (0, 9), and (9, ∞).

Key concepts

x-intercepts

The x-intercepts of a polynomial function are the points where the graph intersects the x-axis. These intercepts are found by setting the polynomial equation equal to zero and solving for the values of x. For example, consider the function \( y = x^3 - 4x^2 - 45x \). To find the x-intercepts, set the equation to zero: \( x^3 - 4x^2 - 45x = 0 \).
Factor out the common term: \( x(x^2 - 4x - 45) = 0 \).
Further factorize to get: \( x(x - 9)(x + 5) = 0 \).
The solutions to this equation are \( x = 0, x = 9, \) and \( x = -5 \).
Therefore, the x-intercepts are \( (0, 0) \), \( (9, 0) \), and \( (-5, 0) \).

degree and end behavior

The degree of a polynomial function is the highest power of \( x \) in the expression. The end behavior of the graph describes how the function behaves as \( x \) approaches infinity (\( \rightarrow \infty \)) or negative infinity (\( \rightarrow -\infty \)).
For example, the function \( y = x^3 - 4x^2 - 45x \) has a degree of 3 because the highest power of \( x \) is 3. Since the leading term is positive \( (x^3) \), as \( x \rightarrow \infty \), \( y \rightarrow \infty \), and as \( x \rightarrow -\infty \), \( y \rightarrow -\infty \).
This means the right side of the graph goes up and the left side goes down.

zeros and multiplicity

Zeros of a polynomial are the values of \( x \) where the function equals zero. Multiplicity refers to the number of times a particular zero appears. To determine multiplicity, factorize the polynomial and observe the exponents of the factors.
For instance, for \( y = x^3 - 4x^2 - 45x \), the factorized form is \( x(x - 9)(x + 5) \).

y-intercept

The y-intercept of a polynomial is the point where the graph intersects the y-axis. To find the y-intercept, substitute \( x = 0 \) into the function and solve for \( y \).
For the function \( y = x^3 - 4x^2 - 45x \), plug in \( x = 0 \):
\( y(0) = 0^3 - 4(0)^2 - 45(0) \)
This simplifies to \( y = 0 \).
Therefore, the y-intercept is \( (0, 0) \).

positive and negative intervals

Positive and negative intervals of a polynomial function describe where the function's values are above or below the x-axis. To find these intervals, consider the zeros found earlier and use a sign chart by testing intervals between these zeros.
For \( y = x^3 - 4x^2 - 45x \), the zeros are \( x = -5 \), \( x = 0 \), and \( x = 9 \).
Test points from each interval:
\( (-\infty, -5) \): Choose \( x = -6 \): \( -6^3 - 4(-6)^2 - 45(-6) > 0 \) - positive
\( (-5, 0) \): Choose \( x = -1 \): \( -1^3 - 4(-1)^2 - 45(-1) < 0 \) - negative
\( (0, 9) \): Choose \( x = 1 \): \( 1^3 - 4(1)^2 - 45(1) < 0 \) - negative
\( (9, \infty) \): Choose \( x = 10 \): \( 10^3 - 4(10)^2 - 45(10) > 0 \) - positive
Thus, the function is positive in the intervals \( (-\infty, -5) \) and \( (9, \infty) \), and negative in the intervals \( (-5, 0) \) and \( (0, 9) \).

y-intercept

The y-intercept of a polynomial is the point where the graph intersects the y-axis. To find the y-intercept, substitute \( x = 0 \) into the function and solve for \( y \).
For the function \( y = x^3 - 4x^2 - 45x \), plug in \( x = 0 \):
\( y(0) = 0^3 - 4(0)^2 - 45(0) \)
This simplifies to \( y = 0 \).
Therefore, the y-intercept is \( (0, 0) \).

positive and negative intervals

Positive and negative intervals of a polynomial function describe where the function's values are above or below the x-axis. To find these intervals, consider the zeros found earlier and use a sign chart by testing intervals between these zeros.
For \( y = x^3 - 4x^2 - 45x \), the zeros are \( x = -5 \), \( x = 0 \), and \( x = 9 \).
Test points from each interval:
\( (-\infty, -5) \): Choose \( x = -6 \): \( -6^3 - 4(-6)^2 - 45(-6) > 0 \) - positive
\( (-5, 0) \): Choose \( x = -1 \): \( -1^3 - 4(-1)^2 - 45(-1) < 0 \) - negative
\( (0, 9) \): Choose \( x = 1 \): \( 1^3 - 4(1)^2 - 45(1) < 0 \) - negative
\( (9, \infty) \): Choose \( x = 10 \): \( 10^3 - 4(10)^2 - 45(10) > 0 \) - positive
Thus, the function is positive in the intervals \( (-\infty, -5) \) and \( (9, \infty) \), and negative in the intervals \( (-5, 0) \) and \( (0, 9) \).