Problem 49 - Precalculus with Calculus Previews

Question

Find the value of \(k\) such that \((0,10)\) is the \(y\) intercept for the graph of \(f(x)=x^{3}-2 x^{2}+14 x-3 k\).

Step-by-Step Solution

Answer

The value of k is \(\frac{-10}{3}\).

Step 1: Understanding the Y-Intercept

The y-intercept of a function is where the graph of the function crosses the y-axis. This occurs when the value of the input (x) is zero.

Step 2: Substitute x with 0 in f(x)

To find the y-intercept substitute x = 0 in the function: \[ f(0) = 0^3 - 2(0)^2 + 14(0) - 3k \] Simplifying this gives: \[ f(0) = -3k \]

Step 3: Set f(0) Equal to 10

Since (0,10) is given as the y-intercept, set the result from the previous step equal to 10: \[ -3k = 10 \]

Step 4: Solve for k

To find the value of k, solve the equation for k: \[ k = \frac{-10}{3} \].

Key Concepts

Cubic Functions

Cubic functions are a type of polynomial function with a degree of three. The general form of a cubic function is expressed as \( f(x) = ax^3 + bx^2 + cx + d \), where \(a\), \(b\), \(c\), and \(d\) are constants and \(a eq 0\). These functions are characterized by their distinctive 'S'-shaped curve, which can take different forms depending on the values of the coefficients.

• The highest degree term, \(x^3\), determines the cubic nature of the function, making it one of the more complex polynomial types.
• Cubic functions can intersect the x-axis up to three times, which indicates they can have one, two, or three real roots.
• The curves of cubic functions can exhibit one or two turning points, resulting in different types of symmetry.

The versatility of cubic functions means they are essential in many applications, from physics to engineering, due to their ability to accurately model certain real-world scenarios.

Y-Intercept

The y-intercept of a polynomial function is a crucial point where the curve graphically crosses the y-axis. This intersection happens when the input \(x\) equals zero. The y-intercept is significant because it provides one of the most straightforward points to plot on a graph and helps in understanding how the graph of the function is positioned relative to the y-axis.

• The y-intercept is given by the value of \(f(0)\), which simplifies the process of finding where the graph intersects the y-axis. In general, for a polynomial function \( f(x) = ax^n + bx^{n-1} + \, ... \, + d \), the y-intercept is simply \(d\).
• For the specific function \(f(x) = x^3 - 2x^2 + 14x - 3k\), substituting \(x = 0\) yields \(f(0) = -3k\), which becomes key in locating the y-intercept when it is known to be \((0,10)\).

Understanding the y-intercept helps in constructing the graph quickly and is often used to solve problems involving the graphical behavior of polynomial functions.

Solving Equations

Solving equations generally involves finding the value of unknown variables that make the equation true. This can include simple linear equations, quadratic equations, and higher-degree polynomials like cubic equations.

• For some polynomial functions, such as cubic functions, solving for unknowns can initially involve substitution and then setting up simpler equations to find unknown variables.
• In scenarios like the given exercise, the process involves solving for \(k\) using a straightforward algebraic manipulation. After calculating \(f(0) = -3k\), the known y-intercept allows us to set \(-3k = 10\) to find \(k\).
• Further solving involves isolating \(k\) by dividing both sides by \(-3\), giving \(k = \frac{-10}{3}\).

This example reinforces how algebraic operations are instrumental in extracting unknown values, which forms a foundation for tackling more complex mathematical tasks.