Problem 28 - Algebra 2

Question

Write an equation for each parabola described below. Then draw the graph. vertex \((0,1),\) focus (0,5)

Step-by-Step Solution

Answer

The equation of the parabola is \( y = \frac{1}{16}x^2 + 1 \). It opens upwards with vertex \((0,1)\) and focus \((0,5)\).

Step 1: Understand the Parabola Structure

A parabola can be described using the formula \[ y = a(x - h)^2 + k \] where \((h, k)\) is the vertex. In this problem, the vertex is \((0,1)\) which indicates that \(h = 0\) and \(k = 1\).

Step 2: Determine the Direction of the Parabola

The parabola opens in the direction of the focus from the vertex. Here, the vertex is at \((0,1)\) and the focus is at \((0,5)\), which is above the vertex. This indicates that the parabola opens upwards.

Step 3: Calculate the Distance (p) from Vertex to Focus

The distance from the vertex \((0,1)\) to the focus \((0,5)\) is \[ p = 5 - 1 = 4. \]This value of \(p\) determines how 'wide' or 'narrow' the parabola is.

Step 4: Find the Value of 'a'

The formula relating \(p\) and \(a\) for a parabola that opens upwards is \[ a = \frac{1}{4p}. \] Substituting \(p = 4\),\[ a = \frac{1}{4(4)} = \frac{1}{16}. \]

Step 5: Write the Equation of the Parabola

Now substitute \(a\), \(h\), and \(k\) into the parabola formula.\[ y = \frac{1}{16}(x - 0)^2 + 1 \] Simplifying, \[ y = \frac{1}{16}x^2 + 1. \]

Step 6: Graph the Parabola

To graph the parabola, plot the vertex at \((0,1)\) and the focus at \((0,5)\). The parabola will be symmetric about the \(y\)-axis, opening upwards. Its vertex is at the bottom point \((0,1)\) and it curves upward towards the focus.

Key Concepts

Vertex

The vertex of a parabola is an essential point that defines its shape and position on the graph. It can be considered the parabola's peak or lowest point, depending on whether it opens upward or downward. In the parabola equation, which is generally expressed as \( y = a(x-h)^2 + k \), the vertex is represented by the coordinates \((h, k)\).

• The form \((h, k)\) indicates the vertex's exact position on the Cartesian plane.
• If the parabola opens upward, the vertex is the minimum point. If it opens downward, the vertex is the maximum point.

In our example, the vertex is at \((0, 1)\). This means the parabola starts at \(y = 1\) on the y-axis and shifts neither left nor right, as \(h = 0\). Understanding the vertex allows us to know where the graph of the parabola begins and its initial direction.

Focus

The focus of a parabola is a point that lies on its axis of symmetry, and it plays a crucial role in defining the parabola's curvature. The distance from the vertex to the focus, denoted as \(p\), helps determine how open or closed the parabola appears.

• The closer the focus is to the vertex, the "steeper" or narrower the parabola looks.
• The farther away the focus, the wider or more spread out the parabola becomes.

In the problem example, the focus is at \((0, 5)\), situated directly above the vertex at \((0, 1)\). This tells us that the parabola opens upwards towards the focus. The vertical distance \(p\) from the vertex to the focus is calculated by subtracting the y-coordinate of the vertex from the y-coordinate of the focus, giving \(p = 4\). This value influences the parabola's 'openness.'

Parabola Equation

The equation of a parabola gives us a mathematical way to describe its curve and symmetry. For parabolas opening up or down, we use the formula \( y = a(x-h)^2 + k \). Here, \(a\) determines the direction and the width of the parabola.

• If \(a\) is positive, the parabola opens upwards; if negative, it opens downwards.
• The magnitude of \(a\) (ignoring sign) tells us about its width; a smaller \(|a|\) value means a wider parabola, while a larger \(|a|\) indicates a narrower one.

In this exercise, since \(p = 4\), the formula for \(a\) is \( a = \frac{1}{4p} \), which results in \(a = \frac{1}{16}\). Substituting into the parabola equation, we get \( y = \frac{1}{16}x^2 + 1 \). This information helps us understand the parabola's width and position relative to its vertex.

Graphing Parabolas

Graphing a parabola involves plotting its vertex, focus, and the general shape influenced by \(a\). Understanding how to graph a parabola is crucial for visualizing the information given by its equation. Here’s a simple guide to graph a parabola:

• Start by plotting the vertex. For our parabola, the vertex is at \((0,1)\).
• Next, determine the direction in which the parabola opens. If the focus is above the vertex, as in our example, the parabola opens upwards.
• Calculate and plot points on either side of the vertex to show symmetry. For every point \(x\) from the vertex, compute \(y\) using the equation \( y = \frac{1}{16}x^2 + 1 \).
• Draw a smooth curve that passes through all these points and curves upward.
• Draw the axis of symmetry vertically through the vertex, ensuring both sides of the parabola reflect across this line.

Graphing parabolas helps to visually interpret their equations and better understand the characteristics defined by the vertex and focus.