Problem 113 - Algebra and Trigonometry

Question

Exercises \(113-115\) will help you prepare for the material covered in the next section. a. If \(y=k x^{2},\) find the value of \(k\) using \(x=2\) and \(y=64\) b. Substitute the value for \(k\) into \(y=k x^{2}\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=5\)

Step-by-Step Solution

Answer

The value of \(k\) is 16. The new equation is \(y = 16x^2\). The value of \(y\) when \(x = 5\) is \(y = 400\).

Step 1: Calculate k

To find the value of the constant \(k\), we substitute \(x = 2\) and \(y = 64\) into the equation \(y = kx^2\) and solve for \(k\). That yields \[64 = k \cdot 2^2\]Dividing both sides of this equation by \(2^2\) or 4 gives us \(k = \frac{64}{4} = 16.\)

Step 2: Substitute k into the equation

Now we substitute \(k = 16\) into the equation \[y = kx^2\]This gives us the resulting equation \[y = 16x^2.\]

Step 3: Find y when x = 5

Now we substitute \(x = 5\) into the equation found above \[y = 16x^2\] That equates to \(y = 16 \cdot 5^2 = 16 \cdot 25 = 400.\)

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is given by \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\]. It includes the terms \(a\), \(b\), and \(c\), which are the coefficients of the equation. Using the quadratic formula can help you find the solutions to the equation when factoring is complex or impossible.
Using this method, students can solve any quadratic equation, as every quadratic can be written in the general form. When applying the quadratic formula, it's crucial to correctly identify the coefficients for \(a\), \(b\), and \(c\) and substitute them into the formula with care to avoid errors. The discriminant, \(b^2 - 4ac\), provides insight into the nature of the roots of the equation; if it's positive, there are two real solutions, if zero, one real solution, and if negative, two complex solutions.

Finding the Value of a Constant

When working with algebraic expressions, we often encounter constants—values that do not change. Finding the value of a constant is a fundamental step that involves substituting known values of variables and solving the resulting equation.
In the given exercise, you are asked to find the constant \(k\) in the equation \(y = kx^2\). By inserting \(x=2\) and \(y=64\), you can determine the value of \(k\) by isolating it on one side of the equation. This process supports the understanding of how constants affect the shape and position of graphs, in this case, the parabola represented by the equation \(y = kx^2\). Understanding this concept is crucial as it applies to multiple areas of algebra and calculus.

Substitution Method

The substitution method is an algebraic technique often used to solve systems of equations and also to find the value of a variable or constant in an equation. It involves replacing a variable with a known value or another expression.
In our exercise, once you find the value of \(k\), you use substitution to replace \(k\) with 16 in the equation \(y = kx^2\). This gives you a new, specific equation to work with: \(y = 16x^2\). When you further replace \(x\) with another number, like 5, you simplify the equation to find \(y\). Substitution is a fundamental skill in algebra that simplifies complex problems and makes them manageable.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (such as addition, subtraction, multiplication, and division). They do not include an equality sign, unlike equations.
Expressions represent values that can change depending on the variables involved. For example, in the expression \((y = kx^2\)), \(k\) and \(x\) are the variables, and when you change these variables, the value of \(y\) also changes. Mastery of working with algebraic expressions is vital for progressing in mathematics, as they are building blocks for more complex concepts in algebra and beyond. Understanding how to manipulate these expressions is key to solving a wide range of problems.