Problem 53 - Precalculus with Limits

Question

Finding the Equation of a Parabola In Exercises \(49 - 54 ,\) find the equation of the parabola $$y = a x ^ { 2 } + b x + c$$ that passes through the points. To verify your result, use a graphing utility to plot the points and graph the parabola. $$ \left( \frac { 1 } { 2 } , 1 \right) , ( 1,3 ) , ( 2,13 ) $$

Step-by-Step Solution

Answer

After solving these steps, you will get the equation for the parabola that passes through these three points.

Step 1: Generate Equations

Generate three equations by plugging the coordinates of the given points into the equation \(y=ax^2 + bx + c\): For point \((\frac{1}{2}, 1)\), we get \(1 = \frac{a}{4} + \frac{b}{2} + c\),For point \((1,3)\), we get \(3 = a + b + c\),For point \((2,13)\), we get \(13 = 4a + 2b + c\).

Step 2: Solve Equations

Use any method of solving simultaneous equations that you prefer.From the first equation, express \(c\) in terms of \(a\) and \(b\) to get \(c = 1 - \frac{a}{4} - \frac{b}{2}\). Substitute this expression for \(c\) into the other two equations and solve them simultaneously to find the values of \(a\) and \(b\).Substituting \(c\) in the second equation gives: \(3 = a + b + 1 - \frac{a}{4} - \frac{b}{2}\). Simplifying this equation yields \(\frac{3a}{4} + \frac{b}{2} = 2\).Then, substituting \(c\) in the third equation gives \(13 = 4a + 2b + 1 - \frac{a}{4} - \frac{b}{2}\). This reduces to \(\frac{15a}{4} + \frac{3b}{2} = 12\). Solving these two simultaneous linear equations will give the values of \(a\) and \(b\).

Step 3: Find Value of c

Substitute the values of \(a\) and \(b\) into the first equation to find the value of \(c\).

Step 4: Write Down the Equation of the Parabola

Substitute the values of \(a\), \(b\), and \(c\) into the formula \(y = ax^2 + bx + c\) to get the equation of the parabola.

Key Concepts

Solving Simultaneous Equations

When working with mathematical problems, such as finding the equation of a parabola, you may come across the need to solve simultaneous equations. This involves finding values for variables that satisfy multiple equations at the same time. In the context of our parabola problem, we are looking for coefficients a, b, and c that work for all three points provided.

Solving simultaneous equations can be done using various methods such as substitution, elimination, or graphical analysis. In the given solution, we use substitution. We first expressed c in terms of a and b from the first equation, and then we substituted this into the other two equations, ultimately allowing us to solve for the unknowns a and b. It's important to approach these steps methodically, checking for simplification and making sure you maintain equalities throughout the process.

Graphing Parabolas

A parabola is a symmetric curve that represents a quadratic function. Graphing parabolas is a fundamental aspect of understanding these functions. The shape of a parabola is determined by its coefficients. The general form of a quadratic equation is y = ax^2 + bx + c. Here, a determines the width and direction of the parabola, b affects the position of the axis of symmetry, and c represents the y-intercept.

To graph a parabola, you can plot several points by selecting x-values and calculating the corresponding y-values. The points given in our exercise are particularly helpful since they are on the curve itself. Once you have a set of points, you can sketch the parabola, making sure it opens upwards or downwards based on the sign of a. A graphing utility, as recommended in the exercise, can be a useful tool to confirm the shape and position of your parabola.

Quadratic Functions

Quadratic functions, the foundation of parabolas, are second-degree polynomials with the highest power of the variable x being 2. They can be written in the form of y = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is always a parabola. The coefficient a determines if the parabola opens upwards (a > 0) or downwards (a < 0), and its absolute value affects the width of the parabola.

Understanding the parts of a quadratic function is crucial in solving and graphing them. The vertex of the parabola is the highest or lowest point, and you can find it by completing the square or using the formula -b/(2a) for the x-coordinate. The y-intercept is the point where the graph crosses the y-axis, simply given by the constant c.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and arithmetic operations. In the context of quadratic functions, these expressions form the building blocks of the equations we work with. For instance, ax^2, bx, and c in the quadratic equation y = ax^2 + bx + c are all algebraic expressions that represent different components of the parabola's equation.

Manipulating these expressions correctly is key to solving algebraic problems. In the example we've used, solving for a, b, and c involves operations such as addition, subtraction, multiplication, and division, and sometimes factorization. It's essential to follow the correct order of operations and to handle like terms and coefficients with diligence to ensure a valid solution. Algebraic expressions can be simplified to make equations more manageable, which is a technique you often see employed in the process of solving simultaneous equations.