Problem 17 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Find \(c\) so that the curve \(y=3 x^{2}-2 x+c\) passes through the point \((2,4)\).

Step-by-Step Solution

Answer

The value of \(c\) is -4.

Step 1: Identify the given point

The curve should pass through the point \(2, 4\). This means that when \(x = 2\), the value of \(y\) should be 4.

Step 2: Substitute the point into the equation

Substitute \(x = 2\) and \(y = 4\) into the equation \(y = 3 x^2 - 2 x + c\). This gives you the equation: \ 4 = 3(2)^2 - 2(2) + c \.

Step 3: Simplify the equation

Calculate \ 3(2)^2\ and \ -2(2) \ to simplify the equation. \(3(2)^2 = 12\) and \( -2(2) = -4\). This leads to the equation: \ 4 = 12 - 4 + c \.

Step 4: Solve for \(c\)

Combine the constant terms on the right-hand side of the equation to get: \ 4 = 8 + c\. Then isolate \(c\) by subtracting 8 from both sides: \ c = 4 - 8 \ which simplifies to \ c = -4 \.

Key Concepts

Quadratic Functions

A quadratic function is a type of polynomial function with the form y = ax^2 + bx + cHere, 'a', 'b', and 'c' are constants, and 'x' is a variable. Quadratic functions parabolas when graphed. The constant 'a' affects the parabola's direction and width. If 'a' is positive, the parabola opens upwards. If 'a' is negative, it opens downwards. The vertex is a key point representing the maximum or minimum of the function. In our exercise, the quadratic function is y = 3x^2 - 2x + c.

Substitution Method

Substitution method involves replacing variables with known values to simplify and solve equations. In our exercise, we are told that the curve passes through point (2,4). This means when x=2, y=4. We substitute these values into the quadratic function equation to get a new equation with only one variable, 'c'.For instance, substituting x=2 and y=4 into y = 3x^2 - 2x + c gives us 4 = 3(2)^2 - 2(2) + c.

Solving for Constants

Solving for a constant involves finding its value by isolating it on one side of the equation. From our new equation, we have 4 = 12 - 4 + c. We first simplify the right-hand side to get 4 = 8 + c. To find c, we subtract 8 from both sides, resulting in c = 4 - 8, which simplifies to c = -4. This shows that the correct value of 'c' is -4, ensuring the curve passes through the point (2,4).

Simplification Steps

Simplification converts complex equations into simpler ones. In our problem, we started with the expression 4 = 3(2)^2 - 2(2) + c. By calculating the powers and products first, we got 4 = 12 - 4 + c. Then, we combined constants on the right-hand side, reducing the equation to 4 = 8 + c. Finally, isolating 'c' simplified our equation to c = -4. Each step reduces the equation's complexity, making it easier to understand and solve.