Problem 82 - Precalculus

Question

Find \(b\) and \(c\) such that the parabola \(y=x^{2}+b x+c\) goes through the points (2,7) and (-6,7).

Step-by-Step Solution

Answer

The values are \(b = 4\) and \(c = -5\).

Step 1: Substitute Point (2,7) into Parabola Equation

Substitute the point (2,7) into the equation of the parabola, which results in:\[ 7 = 2^2 + 2b + c \]This simplifies to:\[ 7 = 4 + 2b + c \]Subtract 4 from both sides:\[ 3 = 2b + c \]This gives us Equation (1): \[ 2b + c = 3 \].

Step 2: Substitute Point (-6,7) into Parabola Equation

Substitute the point (-6,7) into the equation of the parabola:\[ 7 = (-6)^2 + (-6)b + c \]This simplifies to:\[ 7 = 36 - 6b + c \]Subtract 36 from both sides:\[ -29 = -6b + c \]This gives us Equation (2): \[ -6b + c = -29 \].

Step 3: Solve the System of Equations

We now have two equations:Equation (1): \[ 2b + c = 3 \]Equation (2): \[ -6b + c = -29 \].Subtract Equation (1) from Equation (2) to eliminate \(c\):\[ (-6b + c) - (2b + c) = -29 - 3 \]\[ -8b = -32 \]Solving for \(b\):\[ b = 4 \].

Step 4: Find c using b Value

With \(b = 4\), substitute it back into Equation (1):\[ 2(4) + c = 3 \]This simplifies to:\[ 8 + c = 3 \]Subtract 8 from both sides:\[ c = -5 \].

Key Concepts

Solving Simultaneous Equations

When you're tasked with finding unknowns in equations that share variables, you're likely dealing with simultaneous equations. Simultaneous equations consist of two or more equations with multiple variables. In these equations, each unique equation provides additional constraints.

Here are some quick points to understand this concept:

• Each equation offers a relationship among the variables.
• The goal is to find values for these variables that satisfy all equations involved.
• Typical methods to solve include substitution and elimination.

Solving simultaneous equations often boils down to either eliminating variables or substituting into another equation to find values that work across all given equations. Let's look at an example: in our problem, we came across two equations from two points on a parabola. By using one equation to eliminate a variable and solve for the other, we can then substitute that solution back into an equation to find the second variable.

Vertex Form of a Parabola

A parabola is a U-shaped graph that can open upwards or downwards. Parabolas are commonly described using an equation, and one of the most helpful forms is the vertex form.

The vertex form of a parabola is given as:\[y = a(x-h)^2 + k\]where \((h, k)\) is the vertex, the peak or the bottom point of the parabola.
Here are some things to keep in mind:

• Changing \(h\) and \(k\) moves the parabola across the grid without changing its shape.
• The parameter \(a\) dictates how quickly the parabola opens – a larger \(|a|\) means it's narrower.
• If \(a\) is negative, the parabola opens downwards.

In the given exercise, we used a standard form \(y = x^2 + bx + c\) with specific points. This differs slightly from the vertex form but tells us how parabolas can take different forms based on available information and the goal at hand.

Substitution Method

The substitution method is a commonly used technique for solving simultaneous equations, particularly when one equation is simple enough to express one variable in terms of others.

Here's a basic way to use it:

• Start with two equations that share at least one variable.
• Rearrange one equation to express one variable in terms of another.
• Substitute this expression into the other equation.
• Solve the resulting equation for one variable.
• Use this solution to find the second variable by plugging it back into one of the original equations.

In our exercise solution, after getting two equations from the given points on a parabola, we used the substitution method. By expressing \(b\) or \(c\) in terms of each other and substituting, we successfully found values that worked for both equations. This method is incredibly powerful because it simplifies the process of solving more complex systems of equations.