Problem 29 - Algebra 2

Question

Write an equation for each parabola described below. Then draw the graph. vertex \((8,6),\) focus \((2,6)\)

Step-by-Step Solution

Answer

The equation is \((y - 6)^2 = -24(x - 8)\). The parabola opens leftward at vertex \((8,6)\).

Step 1: Identify the Direction of the Parabola

The given vertex is \((8,6)\) and the focus is \((2,6)\). Since both points have the same y-coordinate and the x-coordinate of the focus is smaller than that of the vertex, the parabola opens horizontally to the left.

Step 2: Determine the Distance Between Vertex and Focus

The distance between the vertex \((8,6)\) and the focus \((2,6)\) is calculated by the formula \( p = \text{vertex x-coordinate} - \text{focus x-coordinate} \). Therefore, \( p = 8 - 2 = 6 \).

Step 3: Write the Equation of the Parabola

The standard form of a horizontally oriented parabola with vertex \((h, k)\) and distance \(p\) from vertex to focus is \((y - k)^2 = 4p(x - h)\). Substituting \(h = 8\), \(k = 6\), and \(p = -6\) (negative since it opens to the left), the equation becomes:\[(y - 6)^2 = 4(-6)(x - 8)\] \[ \Rightarrow (y - 6)^2 = -24(x - 8) \]

Step 4: Draw the Graph of the Parabola

Plot the vertex of the parabola at \((8, 6)\) and the focus at \((2, 6)\). Since the parabola opens to the left and is defined by the equation \((y - 6)^2 = -24(x - 8)\), sketch the curve to show it opens to the left from the vertex, curving around the focus.

Key Concepts

Vertex Form of a Parabola

The vertex form of a parabola is a way to express the equation of a parabola using its vertex. Understanding this form can make it easier to graph parabolas and identify important features. The vertex form is written as:\[ y = a(x-h)^2 + k \]Here,

• \(h\) and \(k\) are the coordinates of the vertex \((h, k)\), the point where the parabola changes direction.
• \(a\) determines the width and the direction of the opening of the parabola. If \(a\) is positive, the parabola opens upwards; if negative, it opens downwards. For horizontally opening parabolas, the roles of \(x\) and \(y\) may switch.

It gives students a straightforward view of where the vertex is positioned, helping them draw accurate graphs. This is particularly useful in problems like the one described, which provide the vertex to establish the fundamental shape of the parabola.

Focus and Directrix

The focus and directrix are essential components of a parabola's geometry. They define how the parabola bends and stretch, serving as its 'control settings'.**Focus**The focus of a parabola is a point located at \((h + p,k)\) for a horizontally opening parabola, or \((h, k + p)\) for vertical ones:

• The given point must lie inside the curve of the parabola, meaning the parabola wraps around it.

**Directrix**The directrix complements the focus; it’s a line that runs parallel to the axis of symmetry—located \(p\) units away from the vertex, on the opposite direction to the focus:

• In this scenario, if the focus is \((2, 6)\) and the vertex is \((8, 6)\), the parabola opens to the left and the directrix will be at \(x = 14\), mirroring the focus’ positioning relative to the vertex.

Together, these elements ensure that every point on the parabola is equidistant from the focus and the directrix, forming the unique and predictable arc of the curve.

Graphing Parabolas

Graphing parabolas entails plotting several key elements such as the vertex, focus, and directrix, followed by sketching the symmetrical curve. **Step-by-Step Process** Identify essential points:

• Start by plotting the vertex, as it serves as the central anchor point of your graph.
• Then mark the focus, ensuring it resides within the curve.
• Determine the directrix by plotting it on the opposite side of the focus compared to the vertex.

**Drawing the Curve**

• Sketch the parabola, starting from the vertex and symmetrically wrapping it around the focus.
• Ensure that as you draw, your parabola keeps the focus in the opposing 'bowl' of the curve, consistently reflecting a path where every point is equidistant from both the focus and directrix.

This understanding fundamentally assists in visualizing how parabolas are graphed based on algebraic expressions provided in vertex-focused formats, allowing simplicity and accuracy in identifying each relevant element.