Problem 38 - Algebra 2

Question

Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation. $$ y+x^{2}=-(8 x+23) $$

Step-by-Step Solution

Answer

The equation is a parabola: \( y = -(x + 4)^2 + 7 \).

Step 1: Rearrange Terms

Start by rearranging all terms on one side of the equation to express it in a recognizable form. The given equation is: \( y + x^2 = -(8x + 23) \).First, distribute the negative on the right side: \( y + x^2 = -8x - 23 \).Now, bring all terms to one side: \( y + x^2 + 8x + 23 = 0 \).

Step 2: Identify the Conic Section

Identify the conic section form of the equation \( y + x^2 + 8x + 23 = 0 \).Notice there is only one squared term \( x^2 \). This characteristic indicates that the graph of the equation is a parabola.

Step 3: Write in Standard Parabola Form

To identify the vertex, convert to standard form. Group the quadratic terms and complete the square.Starting with:\( y + x^2 + 8x = -23 \).Focus on the quadratic expression: \( x^2 + 8x \).To complete the square, add and subtract \( (8/2)^2 = 16 \):\( y + (x^2 + 8x + 16 - 16) = -23 \).Rewrite as:\( y + ((x + 4)^2 - 16) = -23 \).Simplify:\( y + (x + 4)^2 = -23 + 16 \), leading to:\( y = -(x + 4)^2 + 7 \).Hence, the standard form is:\( y = -(x + 4)^2 + 7 \).

Step 4: Graph the Parabola

The standard form \( y = -(x + 4)^2 + 7 \) reveals the vertex (-4, 7) and it opens downward. Plot the vertex on the coordinate plane.Since it opens downward, sketch the parabola by plotting additional points, such as (-5, 6) and (-3, 6) for symmetry around the vertex (-4, 7).

Key Concepts

Standard Form

The standard form of a parabola is crucial for understanding and graphically depicting the equation. For a quadratic equation like a parabola, the standard form is given by: \[ y = a(x-h)^2 + k \]Where:

• \( a \) is a constant that affects the direction and width of the parabola. A positive \( a \) opens upward, while a negative \( a \) opens downward.
• \( h \) and \( k \) are the coordinates of the vertex, \( (h, k) \).

In the original exercise, after rearranging and completing the square, the equation is transformed into the standard form \( y = -(x + 4)^2 + 7 \). The term \( (x+4)^2 \) shows the horizontal shift, and the constant +7 indicates the vertical shift. This representation makes it easy to identify properties like the vertex and direction of the parabola. The standard form is valuable for sketching and analyzing the graph.

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane. These include parabolas, circles, ellipses, and hyperbolas. Each conic section has unique characteristics and equations:

• **Parabolas:** One squared term, usually forms a "U" or "n" shape.
• **Circles:** Two squared terms with the same coefficient.
• **Ellipses:** Two squared terms with different coefficients, the sum of which equals a constant.
• **Hyperbolas:** Two squared terms with different coefficients, the difference of which equals a constant.

In this exercise, the equation \( y + x^2 + 8x + 23 = 0 \) features only one squared term, \( x^2 \), which confirms the graph is a parabola. Thus, it belongs to the family of conic sections characterized by a single squared variable.

Quadratic Equations

A quadratic equation is any equation involving a squared term, typically expressed in the form:\[ ax^2 + bx + c = 0 \]Where \( a \), \( b \), and \( c \) are constants. Quadratics often appear in the form of parabola equations. They can be transformed into their standard form for easier graphing and visualization. In this given problem, we started with \( y + x^2 + 8x + 23 = 0 \). To reveal its quadratic nature, we rewrote it into the standard parabolic form \( y = -(x + 4)^2 + 7 \). Here, the quadratic terms are neatly packaged into \( (x+4)^2 \), indicating the importance of completing the square for simplification and graphing purposes. This manipulation is essential for accurately drawing and understanding the graph.

Graphing

Graphing is an insightful way to understand equations visually and interpret their mathematical meanings in geometric terms. For parabolas, the graph is a symmetrical curve defined by its vertex and direction. To graph the standard form equation \( y = -(x + 4)^2 + 7 \):

• Identify the vertex: (-4, 7)
• Note the direction: because the coefficient of the squared term is negative (\( -1 \)), the parabola opens downward.

Start by plotting the vertex on a coordinate plane. Points to the left (-5, 6) and right (-3, 6) help determine the width and symmetry of the parabola. Sketch the curve, ensuring it opens downward as indicated by the negative coefficient. Graphing makes the equation tangible, aiding in understanding its properties and solutions. Always remember to check for symmetry and center your graph around the vertex for accuracy.