Problem 117 - A Graphical Approach to College Algebra

Question

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. Prove that the parabola with focus \((c, 0)\) and directrix \(x=-c\) has equation \(y^{2}=4 c x\)

Step-by-Step Solution

Answer

Equation: \\(y^2 = 4cx\\), Vertex: (0,0), Domain: \\x \\\geq 0\\, Range: all \\y\\ values.

Step 1: Understanding the Components

To prove the equation of the parabola with given focus and directrix, understand the components. The parabola is defined as the set of all points (x, y) equidistant from the focus \(c, 0\) and the directrix \(x = -c\). The distance from any point \(P(x, y)\) on the parabola to the focus is equal to the distance to the directrix.

Step 2: Equating Distances

Calculate the distance from a point \(P(x, y)\) to the focus \(c, 0\) using the distance formula: \sqrt{(x - c)^2 + y^2}\. The distance from \(P(x, y)\) to the directrix \(x = -c\) is the horizontal distance \|(x + c)\|. Set these two distances equal to each other since they are equal for a parabola: \sqrt{(x - c)^2 + y^2} = |x + c|\.

Step 3: Squaring Both Sides

Eliminate the square root by squaring both sides of the equation: \((x - c)^2 + y^2 = (x + c)^2\). This simplifies to \x^2 - 2cx + c^2 + y^2 = x^2 + 2cx + c^2\.

Step 4: Simplifying the Equation

Cancel out \x^2\ and \c^2\ on both sides of the equation: \y^2 - 2cx = 2cx\. Rearrange terms to isolate \y^2\: \y^2 = 4cx\. This equation \(y^2 = 4cx\) is the standard form of a parabola opening to the right.

Step 5: Vertex, Axis, Domain, and Range

The vertex of the parabola \(y^2 = 4cx\) is \(0, 0\). The axis of symmetry is the x-axis \(y = 0\). The domain, since the parabola opens to the right, is \x \geq 0\. The range is all real numbers for \y\.

Step 6: Graphing the Parabola

To graph the parabola \(y^2 = 4cx\), outline the points that satisfy the equation, starting at the vertex moving to the right with focus at \(c, 0\). The graph is symmetrically distributed along the x-axis with the vertex at the origin. Verify the graph using a graphing calculator to ensure its shape and orientation match \(y^2 = 4cx\).

Key Concepts

Vertex and Axis of Symmetry

In the world of parabolas, the vertex serves as the most significant point. It is the point where the parabola changes direction. For the quadratic equation in standard form, the vertex can be identified with coordinates \[ (h, k) \] where the equation is represented as \[ y = a(x-h)^2 + k \]. For the parabola described by \[ y^2 = 4cx \], where the focus is located at \((c, 0)\) and the directrix is \(x = -c\), the vertex in this case is \[ (0, 0) \]. The axis of symmetry here is the x-axis, or simply expressed as \[ y = 0 \].

• The vertex is the turning point at which the parabola pivots.
• The axis of symmetry ensures the parabola is balanced on each side.

Understanding these concepts is especially important when you're plotting the parabola or analyzing its properties.

Graphing Parabolas

Graphing a parabola involves plotting points that satisfy its equation. For the equation \[ y^2 = 4cx \], begin with the vertex at \[ (0,0) \].From this point, identify the focus and directrix:

• Focus: \((c,0)\)
• Directrix: \(x = -c\)

Plotting involves ensuring each point \[ (x, y) \] on the curve is equidistant from both the focus and directrix. The parabola opens to the right, reflecting its orientation along the x-axis.
You can use various points that lie within the boundary of the focus and directrix to ensure accuracy.
Once plotted manually, you can compare with a graphing tool for confirmation.

• Start with the vertex as your reference point.
• Ensure all plotted points maintain symmetry about the x-axis.
• Use symmetry and direction (opening right) as guides.

Domain and Range

The domain and range of a parabola give insights into the possible values of \(x\) and \(y\). For a parabola like\[ y^2 = 4cx \], the domain is constrained by the equation's orientation.
Because it opens to the right, the domain is all values that make \(x\) non-negative, given by:\[ x \geq 0 \].The range for \(y\) is more flexible. Since the value of \(y\) is determined by \(x\), and could be both positive or negative for each \(x\), the range is all real numbers.

• Domain: \(x \geq 0\), indicating the parabola does not extend to negative \(x\) values.
• Range: All real numbers for \(y\), reflecting that the parabola stretches infinitely in the vertical direction.

Understanding domain and range helps in predicting the parabola's extent and behavior.