Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 9: Problem 39
Graph each of the functions. $$f(x)=2 x^{3}+3$$
Short Answer
Expert verified
Graph is S-shaped crossing y-axis at (0, 3).
Step by step solution
01
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.
02
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.
03
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.
04
Plot Key Points
Choose several values for \(x\), calculate the corresponding \(f(x)\), and plot these points. For instance: \(x = -1, f(-1) = 1\); \(x = 1, f(1) = 5\); \(x = 2, f(2) = 2(8) + 3 = 19\).
05
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.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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:
- \( x = -1 \), \( f(-1) = 2(-1)^3 + 3 = 1 \)
- \( x = 1 \), \( f(1) = 2(1)^3 + 3 = 5 \)
- \( x = 2 \), \( f(2) = 2(2)^3 + 3 = 19 \)
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.
