Problem 59 - 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. $$(1.5,6.25),(0,-2),(-1.5,3.25)$$

Step-by-Step Solution

Answer

The equation is \(y = 3x^2 + x - 2\), verified for all points.

Step 1: Identify the Parabola Equation Form

Since the parabola has a vertical axis, it can be expressed by the general quadratic equation \(y = ax^2 + bx + c\). We need to determine the coefficients \(a\), \(b\), and \(c\) using the given points.

Step 2: Plug in the First Point

Use the point \((1.5, 6.25)\) in the equation \(y = ax^2 + bx + c\). This gives us:\[6.25 = a(1.5)^2 + b(1.5) + c\]\[6.25 = 2.25a + 1.5b + c\]

Step 3: Plug in the Second Point

Use the point \((0, -2)\) to get another equation. Substitute \(x = 0\) and \(y = -2\) in \(y = ax^2 + bx + c\):\[-2 = a(0)^2 + b(0) + c\]\[-2 = c\]

Step 4: Substitute the Value of c

From Step 3, we know \(c = -2\). Substitute \(c = -2\) in the equations from Steps 2 and 5, simplifying them to:\[6.25 = 2.25a + 1.5b - 2\]\[8.25 = 2.25a + 1.5b\]

Step 5: Plug in the Third Point

Use the point \((-1.5, 3.25)\) to form another equation. Substitute \(x = -1.5\) and \(y = 3.25\):\[3.25 = a(-1.5)^2 + b(-1.5) + c\]\[3.25 = 2.25a - 1.5b - 2\]\[5.25 = 2.25a - 1.5b\]

Step 6: Solve the System of Equations

We now have the system of equations:\[8.25 = 2.25a + 1.5b\]\[5.25 = 2.25a - 1.5b\]Add the two equations:\[13.5 = 4.5a\]Solve for \(a\):\[a = 3\] Substitute \(a = 3\) into \(8.25 = 2.25a + 1.5b\):\[8.25 = 2.25(3) + 1.5b\]\[8.25 = 6.75 + 1.5b\]\[1.5 = 1.5b\]\[b = 1\]

Step 7: Write the Final Equation

Now that we have \(a = 3\), \(b = 1\), and \(c = -2\), we can write the equation of the parabola as:\[y = 3x^2 + x - 2\]

Step 8: Verification

Verify that the equation \(y = 3x^2 + x - 2\) passes through all given points by substituting \((1.5, 6.25)\), \((0, -2)\), and \((-1.5, 3.25)\). You will find that the points satisfy the equation, confirming its validity.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that can be generally expressed in the form \( y = ax^2 + bx + c \). This is known as the standard form of a quadratic equation. Here’s what each component means:

• \( a \): The coefficient of \( x^2 \) indicates the parabola's degree of curvature. A non-zero \( a \) value means the equation will indeed form a parabola.
• \( b \): This coefficient controls the parabola’s linear component, affecting its direction and horizontal placement.
• \( c \): The constant term \( c \) is the y-intercept, which is where the parabola crosses the vertical axis.

A quadratic equation describes a parabola, a symmetric open curve. Parabolas can open upwards or downwards, depending on the sign of \( a \). In this exercise, the equation was used to find a parabola with a vertical axis that passes through specific data points.

Vertical Axis

The vertical axis is a crucial component in the study of parabolas. When a parabola has a vertical axis, it opens either upwards or downwards, indicating that it is symmetrical about a vertical line. This axis serves as the parabola's line of symmetry. In simple terms, if you folded the parabola along the vertical axis, both halves would match perfectly. For quadratic equations in the form \( y = ax^2 + bx + c \), the vertical axis is generally denoted by the line \( x = -\frac{b}{2a} \). This line is where the parabola will have its vertex, meaning the highest or lowest point, depending on how the parabola opens. Understanding the vertical axis helps when plotting parabolas as it allows prediction of the parabola’s shape and the direction in which it will curve.

System of Equations

A system of equations is a set of two or more equations with the same variables. Solving a system of equations involves finding values for the variables that satisfy all equations simultaneously. In this exercise, we were tasked to find the equation of a parabola using the system of equations derived from substituting the given points into the general quadratic form \( y = ax^2 + bx + c \).

• Each data point provides one equation when substituted into the form.
• Combining these individual equations creates a system of equations.

To solve the system, the equations are typically manipulated through methods like substitution, elimination, or matrix operations to find the coefficients \(a\), \(b\), and \(c\). By solving the system, we ensure that the resulting quadratic equation satisfies each given point, leading to a correct representation of the parabola's characteristics. This step-by-step approach highlights how systems of equations are applied to determine the parameters of quadratic functions.