Problem 36 - Precalculus: Functions and Graphs

Question

If \(f(x)=k x^{3}+x^{2}-k x+2,\) find a number \(k\) such that the graph of \(f\) contains the point \((2,12)\)

Step-by-Step Solution

Answer

The value of \(k\) is 1.

Step 1: Understand the Problem

The problem asks us to find the value of the constant \(k\) such that the function \(f(x) = kx^{3} + x^{2} - kx + 2\) passes through the point \((2, 12)\). This means when \(x = 2\), \(f(x)\) should be 12.

Step 2: Substitute the Point into the Function

Substitute \(x = 2\) and \(f(x) = 12\) into the function \(f(x) = kx^{3} + x^{2} - kx + 2\). This gives us the equation: \[ 12 = k(2)^{3} + (2)^{2} - k(2) + 2 \] Simplifying, we get: \[ 12 = 8k + 4 - 2k + 2 \].

Step 3: Simplify the Equation

Combine like terms in the equation from Step 2: \[ 12 = 8k - 2k + 4 + 2 \]This simplifies to: \[ 12 = 6k + 6 \].

Step 4: Solve for k

Isolate \(k\) by first subtracting 6 from both sides: \[ 12 - 6 = 6k \]This gives us: \[ 6 = 6k \].Finally, divide both sides by 6 to solve for \(k\): \[ k = 1 \].

Key Concepts

Solving Polynomial Equations

Polynomial equations are mathematical expressions that include variables raised to whole number powers and coefficients. Solving these equations involves finding the values of the variable(s) that make the equation true. In this exercise, we want to determine a particular value of the variable \( k \). To do this, we utilize the information given about the point \((2, 12)\) that lies on the graph of the function.
What we do first is substitute the point into the polynomial to create an equation. This means wherever we see \( x \) in the function, we plug in 2 and set the function equal to 12, since that's the function's output at that point. This results in the equation:

• \( 12 = k(2)^3 + (2)^2 - k(2) + 2 \)

The next steps involve simplifying this equation by performing arithmetic operations like multiplication, addition, and combining like terms.
This step-by-step approach is essential. It ensures that we arrive at a clean and tidy simpler equation. Here, after simplification, we get \( 6 = 6k \). Solving for \( k \) involves isolating it, which is quite straightforward as it's already in a simple form.

Function Graph

A function graph represents all the points \((x, f(x))\) that satisfy a function equation. The graph provides a visual picture of how the function behaves over a range of \( x \) values. With polynomial functions, these graphs can take different shapes like lines, parabolas, or more complex curves.

• The degree of the polynomial largely determines the graph's shape. For example, a cubic function often has a curve similar to an 'S' shape.
  Other features include intercepts and turning points.
• The exercise provides a hint by giving a specific point \((2, 12)\) on the graph. It shows where the graph touches a particular area on the coordinate plane.

Understanding graph concepts helps when interpreting polynomial functions because seeing a graph can clarify how changes in coefficients affect the overall shape and position on a plane. Interpreting this lets us predict function behavior without calculating every single value.

Function Substitution

Function substitution is a mathematical technique used to evaluate functions at specific points or to manipulate them for easier problem-solving. By replacing \( x \) with a specific value, we simplify the function to a numerical expression, which then reveals specific characteristics of the polynomial.

• In our exercise, substitution is key because we're given \( x = 2 \) and need \( f(x) = 12 \) to solve for \( k \).

Think of function substitution as "testing" a point within the function. By checking this value, we confirm if it aligns with the given conditions of the problem -- if not, it indicates an adjustment needed somewhere in the equation.
This tactic not only solves equations, but it also serves to verify solutions since plugging the outcome back into the original expression should maintain equality, effectively confirming the correctness.