Question

Graph each polynomial function. $$ f(x)=x^{4}-3 x^{2}-4 $$

Short answer

The roots are \( x = 2 \) and \( x = -2 \). The y-intercept is at \( (0, -4) \). The polynomial has end behavior pointing upwards at both ends.

Step-by-step solution

- Identify the Degree and Leading Coefficient

The given polynomial function is \( f(x) = x^4 - 3x^2 - 4 \). Identify the degree and the leading coefficient. The degree of the polynomial is 4, and the leading coefficient (coefficient of the term with the highest power) is 1.

- Determine the End Behavior

Since the degree of the polynomial is 4 (even) and the leading coefficient is positive, both ends of the graph will point upwards. This means that as \( x \to \, \, \pm \, \, \infty \), \( f(x) \to \, \, \infty \).

- Find the Roots/Intercepts

To find the roots of the polynomial, solve \( f(x) = 0 \):\[ x^4 - 3x^2 - 4 = 0 \]. Let \( u = x^2 \), so the equation becomes \( u^2 - 3u - 4 = 0 \). Factor the quadratic equation: \[ u^2 - 3u - 4 = (u - 4)(u + 1) = 0 \]. Thus, \( u = 4 \) or \( u = -1 \). Since \( u = x^2 \), we have \( x^2 = 4 \) (so \( x = \pm 2 \)) and \( x^2 = -1 \) (which has no real solution). The roots/intercepts are \( x = 2 \) and \( x = -2 \).

- Determine the Y-Intercept

To find the y-intercept, evaluate \( f(0) \): \[ f(0) = 0^4 - 3(0)^2 - 4 = -4 \]. The y-intercept is at the point \( (0, -4) \).

- Analyze Critical Points and Intervals

Find the first derivative of \( f(x) \) to determine critical points: \[ f'(x) = 4x^3 - 6x \]. Set \( f'(x) = 0 \): \[ 4x^3 - 6x = 0 \]. Factor out \( 2x \): \[ 2x(2x^2 - 3) = 0 \]. This gives \( x = 0 \) or \( 2x^2 - 3 = 0 \). Solving \( 2x^2 - 3 = 0 \) provides \( x = \pm\frac{\root{3}}{2} \). Evaluate \( f(x) \) at these points to analyze function behavior in different intervals.

- Sketch the Graph

Combine all the information: the polynomial's end behavior, roots, y-intercept, and analysis of critical points. Plot these points and draw a smooth curve through them considering the end behavior. Label the key points \( (-2, 0) \), \( (2, 0) \), and \( (0, -4) \).

Key concepts

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial. For the function \( f(x) = x^4 - 3x^2 - 4 \), the degree is 4 since the highest power of \( x \) is 4. Understanding the degree is crucial because it tells us about the shape and complexity of the graph. A polynomial of degree 4 can have up to 4 roots and 3 turning points.

Leading Coefficient

The leading coefficient is the coefficient of the term with the highest power in a polynomial. In \( f(x) = x^4 - 3x^2 - 4 \), the leading coefficient is 1. The leading coefficient affects the end behavior of the graph. For instance, if it's positive, the polynomial's ends point upwards; if it's negative, they point downwards.

End Behavior

End behavior describes what happens to the value of \( f(x) \) as \( x \) approaches positive or negative infinity. For the polynomial \( f(x) = x^4 - 3x^2 - 4 \), since the degree is even (4) and the leading coefficient is positive (1), both ends of the graph go upwards. Mathematically,

• As \( x \to +fty, f(x) \to +fty \)
• As \( x \to -fty, f(x) \to +fty \)

Roots of Polynomial

Roots of the polynomial are the values of \( x \) where \( f(x) = 0 \). For \( f(x) = x^4 - 3x^2 - 4 \), set \( f(x) = 0 \) and solve: Let \( u = x^2 \), transforming the equation to \( u^2 - 3u - 4 = 0 \). Factoring, we get \((u-4)(u+1) = 0 \), so \( u = 4 \) or \( u = -1 \). Thus, \( x^2 = 4 \) (yielding \( x = \pm 2 \)) and \( x^2 = -1\) (which has no real solution). Therefore, the real roots are \( x = 2 \) and \( x = -2 \).

Y-Intercept

The y-intercept is where the graph crosses the y-axis, found by evaluating \( f(0) \). For \( f(x) = x^4 - 3x^2 - 4 \), we compute \( f(0) = 0^4 - 3(0)^2 - 4 = -4 \). Thus, the y-intercept is at \( (0, -4) \).

Critical Points

Critical points are where the derivative equals zero or is undefined, indicating potential maxima, minima, or inflection points. To find them, take the first derivative of \( f(x) = x^4 - 3x^2 - 4 \), giving us \( f'(x) = 4x^3 - 6x \). Setting \( f'(x) = 0 \), we factor to get \( 2x(2x^2 - 3) = 0 \), resulting in \( x = 0 \) or \( x = \pm \sqrt{3/2} \). These points help analyze the function’s behavior.

Derivative

The derivative represents the rate of change of the function. For \( f(x) = x^4 - 3x^2 - 4 \), the first derivative \( f'(x) = 4x^3 - 6x \) helps find critical points and understand the function's behavior. Solving \(4x^3 - 6x = 0 \) by factoring yields critical points. Evaluating \( f(x) \) at these points identifies intervals of increase or decrease and potential maxima or minima.