Problem 39 - Intermediate Algebra

Question

Graph each of the functions. $$f(x)=2 x^{3}+3$$

Step-by-Step Solution

Answer

Graph is S-shaped crossing y-axis at (0, 3).

Step 1: Identify the Function

The function given is \(f(x) = 2x^3 + 3\). This is a cubic function due to the highest power of \(x\) being 3.

Step 2: Understand the Basic Shape

A cubic function generally has an S-like curve. The term \(2x^3\) affects how stretched or compressed the graph is compared to the basic \(x^3\) graph.

Step 3: Determine Intercepts

To find the y-intercept, substitute \(x = 0\): \(f(0) = 2(0)^3 + 3 = 3\). Thus, the y-intercept is at \((0, 3)\). Since this is a cubic function, there is no simple x-intercept without solving the equation \(2x^3 + 3 = 0\), which is complex and will require more advanced methods.

Step 4: Plot Key Points

Choose several values for \(x\), calculate the corresponding \(f(x)\), and plot these points. For instance: \(x = -1, f(-1) = 1\); \(x = 1, f(1) = 5\); \(x = 2, f(2) = 2(8) + 3 = 19\).

Step 5: Sketch the Graph

Using the origin of the characteristic cubic curve, the calculated points, and the general direction as mapped out from the points, draw a smooth S-shaped curve crossing the y-axis at (0, 3). The curve will decrease towards negative infinity as \(x\) goes towards negative infinity and will rise towards positive infinity as \(x\) increases beyond zero.

Key Concepts

Cubic Function

A cubic function is a type of polynomial function where the highest degree, or power, of the variable is three. This is reflected in its standard form, which looks like this: \[ f(x) = ax^3 + bx^2 + cx + d \] In this function:

• \( a \), \( b \), \( c \), and \( d \) are constants and real numbers
• \( a eq 0 \) since \( a = 0 \) would turn the equation into a quadratic function, not cubic.

The function \( f(x) = 2x^3 + 3 \) is indeed a cubic function as its highest degree is 3, which is signified by the term \( 2x^3 \). This particular function includes only one variable term with \( x^3 \), meaning it adjusts the basic cubic shape, expanding or compressing it by the factor of 2.

Y-Intercept

A y-intercept is where your graph crosses the y-axis. This point can be found by calculating the value of the function when \( x = 0 \). For cubic functions like \( f(x) = 2x^3 + 3 \), you substitute zero for every \( x \), solving the equation for \( f(0) \): \[ f(0) = 2(0)^3 + 3 = 3 \] This means the y-intercept here is at the point \( (0, 3) \). On the graph, this shows the specific location where your curve intersects the vertical y-axis.

• It is essential because it gives you a starting point in sketching the graph.
• Finding the y-intercept is usually more straightforward than finding the x-intercept in cubic functions.

Plotting Points

To successfully graph a cubic function, you'll need several points plotted on the coordinate plane. This involves choosing several values for \( x \) and calculating their respective \( f(x) \) values. Consider these points based on the function \( f(x) = 2x^3 + 3 \), for instance:

• \( x = -1 \), \( f(-1) = 2(-1)^3 + 3 = 1 \)
• \( x = 1 \), \( f(1) = 2(1)^3 + 3 = 5 \)
• \( x = 2 \), \( f(2) = 2(2)^3 + 3 = 19 \)

Plot these calculated points on the graph, using their respective \( x, y \) coordinates. These points give a foundation for the curve and guide how to sketch the overall graph, showing the changes in its direction and ascent or descent.

S-Shaped Curve

The graph of a cubic function typically forms an S-shaped curve. This feature differentiates it from other polynomial graphs like linear or quadratic functions. Understanding its S-shape can help visualize how it behaves on a coordinate plane.

• The curve will usually start increasing or decreasing, depending on the sign and magnitude of \( a \) in the \( ax^3 \) term.
• As \( x \) moves towards positive infinity, the graph tends to rise into positive infinity if \( a \) is positive.
• When \( x \) approaches negative infinity, the curve typically further decreases into negative infinity if \( a \) is positive.
• This curve provides a visual indication of how solutions (or roots) behave, even when imaginary or complex roots are involved.