Problem 70 - Intermediate Algebra

Question

Write each equation in standard form, if it is not already so, and graph it. If the graph is a circle, give the coordinates of its center and its radius. If the graph is a parabola, give the coordinates of its vertex. $$ x=-y^{2}-5 $$

Step-by-Step Solution

Answer

The graph is a parabola; vertex \((-5, 0)\); opens left.

Step 1: Identify the equation type

The equation given is \( x = -y^2 - 5 \). This equation has a \( y^2 \) term and no \( x^2 \) term, indicating that it is a parabola. Parabolas can open vertically or horizontally, and since the \( y \) term is squared, this parabola opens left or right.

Step 2: Rewrite equation in standard form

For parabolas that open horizontally, the standard form is \((y-k)^2 = 4p(x-h)\), where \((h, k)\) is the vertex and \(p\) determines the direction and width of the parabola. To rewrite \( x = -y^2 - 5 \), add 5 to both sides to get \( x+5 = -y^2 \). Thus, \( -(y-0)^2 = x+5 \).

Step 3: Convert to align with horizontal parabola form

Recognize that \( -(y-0)^2 = (x+5) \) can be expressed as \( (y-0)^2 = -1(x - (-5)) \). This matches the form \((y-k)^2 = 4p(x-h)\) with \( k = 0 \), \( h = -5 \), and \( 4p = -1 \). Thus, \( p = -\frac{1}{4} \).

Step 4: Determine vertex and direction

From the standard form \((y - k)^2 = 4p(x - h)\), we identify that the vertex \((h, k)\) is \((-5, 0)\). Since \( p = -\frac{1}{4} \), the parabola opens to the left.

Step 5: Graph the equation

To graph the equation, plot the vertex at \((-5, 0)\), then use the value of \( p \) to determine that the parabola opens leftward as it expands. Use additional points to guide the curvature, such as choosing y-values like 1 and 2 to find corresponding x-values and plot these.

Key Concepts

Vertex Identification

Identifying the vertex of a parabola is a key step when working with quadratic equations. The vertex is a critical point and often represents the minimum or maximum point on the graph. In the case of horizontal parabolas like the given equation \[ x = -y^2 - 5 \], the equation can be rewritten in the standard form as \[(y-k)^2 = 4p(x-h)\]. This rewrite allows us to easily identify the vertex.

• For the equation \[ -(y-0)^2 = x+5 \], we have \[ (y-0)^2 = -1(x - (-5)) \].
• Here, the vertex, \( (h, k) \), is easily spotted as \((-5, 0)\).

The vertex represents the point where the parabola turns, enabling us to understand its orientation and position on the graph. For parabola equations in the form \[(y-k)^2 = 4p(x-h)\], the vertex is always at \((h, k)\). "h" and "k" shift the graph horizontally and vertically, respectively.

Graphing Parabolas

Graphing parabolas might seem challenging, but breaking the process into steps simplifies it. Once you have the equation in standard form, you’ll know where to start your graph. With the standard form \((y - k)^2 = 4p(x - h)\),identifying the direction in which the parabola opens becomes easier.

• The vertex \((-5, 0)\) provides a pivotal point from which to plot the graph.
• Understanding "p" is crucial: a positive p means the parabola opens to the right, while a negative p indicates it opens to the left.In this case, p = -\frac{1}{4}\, needing our parabola to open leftward.

Plotting additional points clarifies the parabola’s shape. Pick y-values, substitute them back into the equation to find corresponding x-values, and plot these points. This ensures your graph reflects the parabola's correct orientation and curvature.

Standard Form of Parabolas

The standard form of a parabola equation is transformative for graphing and further analysis. For parabolas that open horizontally, the form is \((y-k)^2 = 4p(x-h)\). Here, each component represents:

• "k" and "h" mark the vertical and horizontal shifts, identifying the vertex \((h, k)\).
• "p" impacts the direction and width; negative values open left or down, while positive values open right or up.

When given the equation \[ x = -y^2 - 5 \], you convert it to the standard form\[ (y-0)^2 = -1(x - (-5)) \],making it clearer that:

• The vertex is \((-5, 0)\)
• "4p" equals \(-1\), which implies \(p = -\frac{1}{4}\), signifying a leftward opening.

Utilize standard form conversion for accurate graphing and deeper understanding of the parabola's properties.