Problem 47 - Algebra 2

Question

Write an equation for a parabola with vertex at \((-3,-4)\) and \(y\) -intercept 8

Step-by-Step Solution

Answer

The equation of the parabola is \(y = \frac{4}{3}(x + 3)^2 - 4\).

Step 1: Understand the Parabola Equation

A parabola with a vertical axis of symmetry can be described by the equation \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola.

Step 2: Substitute the Vertex into the Equation

Given that the vertex of the parabola is \((-3, -4)\), we substitute \(h = -3\) and \(k = -4\) into the equation: \[ y = a(x + 3)^2 - 4 \]

Step 3: Use the Y-Intercept to Find the Coefficient a

The \(y\)-intercept is the value of \(y\) when \(x = 0\). Since the \(y\)-intercept is \(8\), substitute \(x = 0\) and \(y = 8\) into the equation: \[ 8 = a(0 + 3)^2 - 4 \] The equation simplifies to: \[ 8 = 9a - 4 \]

Step 4: Solve for a

To find \(a\), solve the equation from Step 3: 1. Add 4 to both sides: \[ 12 = 9a \] 2. Divide both sides by 9: \[ a = \frac{12}{9} = \frac{4}{3} \]

Step 5: Write the Equation

Substitute the value of \(a\) back into the parabola's equation: \[ y = \frac{4}{3}(x + 3)^2 - 4 \] This is the equation of the parabola with the specified vertex and \(y\)-intercept.

Key Concepts

Vertex Form

The vertex form of the equation of a parabola is an essential tool when studying quadratic functions. This form can be written as \( y = a(x - h)^2 + k \), where \((h, k)\) represents the vertex of the parabola.
The vertex is simply the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards. In this equation:

• \(a\) determines the direction and width of the parabola.
• \(h\) and \(k\) indicate the coordinates of the vertex.

Using the vertex form makes it straightforward to identify the vertex of the parabola by simply observing the equation. It's particularly useful for graphing, allowing you to easily sketch the basic shape of the parabola.

Solving for a

In the context of a parabola, the coefficient \(a\) in the vertex form equation \(y = a(x - h)^2 + k\) plays a vital role in defining the parabola's shape. When constructing a specific parabola, as in the given exercise, finding the value of \(a\) is crucial.
To solve for \(a\), you need another point on the parabola in addition to the vertex. Often, the y-intercept is used for this purpose because it's straightforward to find. Here's a quick way to solve for \(a\):

• Identify the known points, such as the vertex and y-intercept.
• Substitute these values into the vertex form equation to create an equation.
• Solve this equation for \(a\).

In our example, we used the y-intercept \( (0, 8) \) to find \( a = \frac{4}{3} \). The sign and size of \(a\) tell us if the parabola opens upwards or downwards and how "stretched" or "compressed" it looks.

Parabola Features

Parabolas have key features that define their shape and position on a graph. Understanding these features helps in graphing and analyzing quadratic functions, like the one in our exercise.
Here are the essential features to consider:

• Vertex: The turning point of the parabola, where it reaches its maximum or minimum.
• Axis of Symmetry: A vertical line through the vertex, splitting the parabola into two mirror-image halves.
• Direction: Determined by \(a\), the parabola can open upwards (if \(a > 0\)) or downwards (if \(a < 0\)).
• Width: The value of \(a\) also affects the width; smaller \(|a|\) values result in a wider parabola, while larger \(|a|\) values produce a narrower one.

Grasping these features allows you to accurately describe and plot the shape and position of any parabola on the coordinate plane.

Y-Intercept in Parabolas

The y-intercept of a parabola is where it crosses the y-axis, giving a point with coordinates \((0, y)\). It's an essential point because it offers a real value to work with when deriving the equation of the parabola from its vertex form.
To find the y-intercept in a parabola equation:

• Set \(x = 0\) in the equation \(y = a(x - h)^2 + k\).
• Simplify to find \(y\).

The y-intercept is particularly useful for identifying the equation of a parabola, especially when combined with information about the vertex. In example cases like this one, knowing the y-intercept \((0, 8)\) was instrumental in solving for \(a\) and finalizing the equation of the parabola: \( y = \frac{4}{3}(x + 3)^2 - 4 \). Understanding this point on the parabola can greatly assist in constructing the complete graph of the function.