Problem 19 - College Algebra Essentials

Question

In Exercises 19-22, find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. $$(-1,6),(1,4),(2,9)$$

Step-by-Step Solution

Answer

The quadratic function that passes through the points (-1,6), (1,4), and (2,9) is \(y=-x^{2}+2.5x+2.5\).

Step 1: Formulate the equations

Insert the points into the quadratic function: From \((-1,6)\), the equation is \(a-b+c=6\). From \((1,4)\), the equation is \(a+b+c=4\). From \((2,9)\), the equation is \(4a+2b+c=9\)

Step 2: Solve the system of equations

Use the three equations from Step 1 to solve for \(a\), \(b\) and \(c\). Subtract the second equation from the first to eliminate \(b\) and obtain \(a=-1\).Substitute \(a=-1\) into the second equation to find \(b=2.5\) and substitute \(a=-1\) and \(b=2.5\) into the third equation to find \(c=2.5\).

Step 3: Formulate the quadratic function

Combine the values for \(a\), \(b\) and \(c\) obtained in Step 2 to derive the desired quadratic function. \(y=-x^{2}+2.5x+2.5\)

Key Concepts

Solving Systems of Equations

In mathematics, solving a system of equations involves finding values for the variables that satisfy all the given equations. In the context of quadratic functions, like in the current problem, we often deal with three equations as we have three points through which the graph passes. This is because a quadratic function in the standard form, \(y = ax^2 + bx + c\), has three coefficients: \(a\), \(b\), and \(c\), to find.

Here's a step-by-step guide on solving systems of equations for finding these coefficients:

• Identify and substitute the coordinates of the points into the quadratic equation to form a system of equations. For point \((-1, 6)\), plug these into \(a(-1)^2 + b(-1) + c = 6\).
• Use the method of elimination or substitution to solve these equations step by step. First, try to eliminate one variable, making it easier to solve the others.
• Substitution is where you solve one equation for one variable and then substitute this back into the other equations.
• Check your solutions by substituting back into the original equations to ensure they satisfy all given conditions.

For example, eliminating \(b\) from two equations and then solving for \(a\), followed by finding \(b\) and \(c\) is a common approach in this process. This would ensure that you've correctly solved for all the unknowns.

Graph of a Function

The graph of a quadratic function is called a parabola. It can open upwards or downwards, depending on the coefficient \(a\). In the given exercise, once we found \(a = -1\), \(b = 2.5\), and \(c = 2.5\), the quadratic equation becomes \(y = -x^2 + 2.5x + 2.5\).

Here are some key features of the graph of a quadratic function:

• Vertex: The point where the parabola changes direction. It is the peak if the parabola opens downwards (as in this case with a negative \(a\)), or the lowest point if it opens upwards.
• Axis of Symmetry: A vertical line that divides the parabola into two mirror-image halves. For a parabola \(y = ax^2 + bx + c\), the equation for the axis of symmetry is \(x = -\frac{b}{2a}\).
• Intercepts: The points where the parabola crosses the x-axis and y-axis. The y-intercept is found by substituting \(x = 0\), and for x-intercepts, solve \(y = 0\).

Graphing the quadratic function helps in visualizing how it behaves and confirms the solution of an algebraic problem. It's also crucial for checking whether the calculated points actually lie on the graph.

Finding Coefficients

Finding the coefficients \(a\), \(b\), and \(c\) in a quadratic equation is all about identifying the specific numbers that shape the parabola and determine its orientation and width. Using three points, as in this exercise, gives us enough data to uniquely determine these coefficients.

The basic steps include:

• Substitute Known Points: Each point on a graph can be plugged into the quadratic equation to form a separate equation. For example, using point \((1, 4)\) results in the equation \(a(1)^2 + b(1) + c = 4\).
• Solve the System: Once the equations are formed, they can be solved simultaneously to find the exact values of \(a\), \(b\), and \(c\).
• Verify: After calculating, substitute the values back into the original equations to verify correctness.

This method ensures that the quadratic function accurately represents the path of the graph through all given points. By understanding these coefficients, students can derive a quadratic formula from data points and understand the underpinnings of the graph's shape.