Problem 29 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 29-34: Find the standard equation of a parabola that has a vertical axis and satisfies the given conditions. Vertex \((0,-2)\), passing through \((3,25)\)

Step-by-Step Solution

Answer

The equation is \\(y = 3x^2 - 2\\).

Step 1: Understand the Parabola Equation

For a parabola with a vertical axis and the vertex at \(h, k\), the equation is given by \(y = a(x-h)^2 + k\). In this exercise, we have the vertex at \(h = 0\) and \(k = -2\).

Step 2: Substitute Vertex into Equation

Substitute \(h = 0\) and \(k = -2\) into the standard form equation to get \(y = a(x-0)^2 - 2\) or simply \(y = ax^2 - 2\).

Step 3: Use the Given Point (3, 25)

Substitute the point \(x = 3\) and \(y = 25\) into the equation \(y = ax^2 - 2\) to find the value of \(a\).

Step 4: Solve for 'a'

Plug x = 3 and y = 25 into the equation: \((25 = a(3)^2 - 2)\). Simplify the equation to get \(25 = 9a - 2\).

Step 5: Isolate 'a'

Add 2 to both sides: \(27 = 9a\). Then divide by 9: \(a = 3\).

Step 6: Write the Standard Form Equation

Substitute \(a = 3\) back into the equation \(y = ax^2 - 2\). So, the standard equation is \(y = 3x^2 - 2\).

Key Concepts

Vertex Form of a Parabola

The vertex form of a parabola is an essential way to express the equation of a parabola that helps identify its vertex clearly. This form is written as

• \(y = a(x-h)^2 + k\)

Here, \((h, k)\) is the vertex of the parabola, which is a crucial point that gives the parabola its distinct shape and orientation. In this equation:

• \(h\) represents the horizontal offset from the origin.
• \(k\) represents the vertical offset from the origin.
• \(a\) determines the "open-ness" and direction of the parabola. If \(a\) is positive, the parabola opens upwards. If negative, it opens downwards.

Understanding the vertex form makes it easier to graph the parabola and solve for key attributes like the axis of symmetry. Because of its intuitive layout, it is particularly useful for identifying how the parabola shifts on a graph. Remember, any change in \(h\) or \(k\) will change the position of the parabola's vertex on the Cartesian plane.

Vertical Axis

A parabola with a vertical axis means that its axis of symmetry is a vertical line. This is an imaginary line that passes through the vertex, dividing the parabola into two mirror-image halves. For a parabola with a vertical axis, the equation is typically:

• \(y = ax^2 + bx + c\)

This form assumes the parabola is oriented vertically, which is the most common orientation you will encounter in basic algebra.The vertical axis can be visualized as a line \(x = h\) that passes through the vertex \((h,k)\). This means the parabola is symmetric with respect to this line. Knowing the axis of symmetry allows students to better plot the parabola and understand its symmetrical properties. Since each point \((x, y)\) on one side of the axis has a corresponding point \((x', y)\) on the other side, it simplifies graphing and provides a clear picture of the parabola's structure.

Standard Equation of a Parabola

The standard equation of a parabola describes the shape and position of a parabola on the coordinate plane. It is written as

• \(y = ax^2 + bx + c\)

In this equation, the terms have specific roles:

• \(a\) influences the direction and narrowness of the parabola. A positive \(a\) opens the parabola upwards, while a negative \(a\) opens it downwards. Larger values of \(a\) make the parabola narrower, whereas smaller ones make it wider.
• \(b\) determines the position of the vertex horizontally.
• \(c\) dictates the vertical position where the parabola intersects the y-axis.

In parabolas with a vertical axis, the standard equation forms the basis for deriving specific vertex form equations. It allows us to analyze and predict the behavior of quadratic functions based on algebraic calculations, aiding in solving real-world problems with parabolic trajectories.