Problem 62 - A Graphical Approach to College Algebra

Question

Find the equation of the parabola with vertical axis that passes through the data points shown or specified. Check your answer. $$(-2,2),(0,2),(2,-6)$$

Step-by-Step Solution

Answer

The equation is \( y = -x^2 - 2x + 2 \).

Step 1: Understand the Parabola Equation

A parabola with a vertical axis can be represented by the equation \( y = ax^2 + bx + c \). We need to determine the coefficients \(a\), \(b\), and \(c\) using the given points.

Step 2: Set Up the System of Equations

We have three points: \((-2, 2)\), \((0, 2)\), and \((2, -6)\). Substitute these points into the equation \(y = ax^2 + bx + c\) to get three equations:\1. For \((-2, 2)\): \[ 4a - 2b + c = 2 \]2. For \((0, 2)\): \[ c = 2 \]3. For \((2, -6)\): \[ 4a + 2b + c = -6 \]

Step 3: Solve the System of Equations

Substitute \( c = 2 \) from the second equation into the other two equations:\1. \(4a - 2b + 2 = 2\), simplifies to \(4a - 2b = 0\).2. \(4a + 2b + 2 = -6\), simplifies to \(4a + 2b = -8\).

Step 4: Use Elimination or Substitution to Find \(a\) and \(b\)

Using \(4a - 2b = 0\), solve for one of the variables. Dividing the entire equation by 2, we get:\[ 2a - b = 0 \] to give us \( b = 2a \).Substitute \( b = 2a \) into the second simplified equation:\[ 4a + 2(2a) = -8 \] which simplifies to \[ 4a + 4a = -8 \] or \[ 8a = -8 \]. So, \(a = -1 \).

Step 5: Find \(b\) Using \(a\)

Substitute \( a = -1 \) into \( b = 2a \):\[ b = 2(-1) = -2 \].

Step 6: Write Down the Equation of the Parabola

Now that we have \(a\), \(b\), and \(c\), we can write the equation of the parabola:\[ y = -x^2 - 2x + 2 \]

Step 7: Verify the Equation with the Given Points

Plug the original points into the obtained equation to ensure they satisfy:1. For \((-2, 2)\):\[ -(-2)^2 - 2(-2) + 2 = -4 + 4 + 2 = 2 \]2. For \((0, 2)\):\[ -(0)^2 - 2(0) + 2 = 2 \]3. For \((2, -6)\):\[ -(2)^2 - 2(2) + 2 = -4 - 4 + 2 = -6 \]Hence, all points satisfy the equation.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function that has the general form \( y = ax^2 + bx + c \). In this equation, \(a\), \(b\), and \(c\) are constants with \(a eq 0\). The highest power of \(x\) in the equation is 2, which is why it's called a "quadratic" function. These functions graph as parabolas, which are U-shaped curves.

• If \(a\) is positive, the parabola opens upwards.
• If \(a\) is negative, the parabola opens downwards.
• The sign and size of \(b\) and \(c\) affect the width and position of the parabola.

Next, let's understand how these constants impact the graph:

• \(a\) affects the direction and width of the parabola. The greater the absolute value of \(a\), the "narrower" the parabola appears.
• \(b\) influences the horizontal placement and the tilt of the parabola's axis of symmetry.
• \(c\) represents the y-intercept, where the parabola crosses the y-axis.

This function is essential because it models numerous real-world situations, from physics trajectories to economic phenomena.

System of Equations

When dealing with a quadratic function, you might need to find specific coefficients using a set of points. This process often involves setting up a system of equations. A system of equations consists of multiple equations that you solve simultaneously. Each equation provides information about the unknowns based on given conditions.

In the context of quadratic functions, if you have a parabola that must pass through a set of points, you can substitute these points into the equation \( y = ax^2 + bx + c \) to create separate equations for each point. This gives us a system, which typically consists of three equations when three points are involved.

• The first equation comes from the first point substituted in the general parabola formula.
• The second equation comes from the second point. Often, this can simplify a variable if the x-coordinates permit.
• The third equation uses the third point, completing the system.

Solving these simultaneously allows you to find the values of \(a\), \(b\), and \(c\). Techniques such as substitution, elimination, or matrix operations are commonly applied to solve these systems.

Vertical Axis Parabola

A parabola with a vertical axis is one where the axis of symmetry is parallel to the y-axis. This means the parabola opens either upwards or downwards. The equation of a vertical axis parabola has a simple standardized form: \( y = ax^2 + bx + c \).

These parabolas have specific characteristics:

• The vertex lies somewhere on the y-axis.
• The axis of symmetry for the parabola is represented by a vertical line, \(x = -\frac{b}{2a}\).
• All vertical axis parabolas have the same basic shape, differing in width and orientation based on \(a\), \(b\), and \(c\).

Understanding vertical axis parabolas is crucial when modeling phenomena that follow a symmetric path. Such as objects in freefall under gravity, where the path of motion is symmetric with respect to a vertical line.