Problem 35 - Algebra and Trigonometry

Question

\(35-38\) Use a graphing device to graph the conic. $$ 2 x^{2}-4 x+y+5=0 $$

Step-by-Step Solution

Answer

The conic is a downward-opening parabola with vertex at (1, -3).

Step 1: Rewrite the Equation

The given equation is \( 2x^2 - 4x + y + 5 = 0 \). To facilitate graphing, we should rewrite this in standard form for a conic, such as a parabola. Begin by isolating the term with \( y \): \( y = -2x^2 + 4x - 5 \).

Step 2: Complete the Square

To express the quadratic part in a form that reveals its vertex, complete the square on the \( x \) terms. Begin with the coefficient of \( x^2 \), which must be 1, so factor out the 2: \( y = -2(x^2 - 2x) - 5 \). Completing the square inside the parentheses, \( x^2 - 2x \) becomes \( (x - 1)^2 - 1 \): so \( y = -2((x - 1)^2 - 1) - 5 \).

Step 3: Simplify the Equation

Distribute the \(-2\) and simplify: \( y = -2(x - 1)^2 + 2 - 5 \), which simplifies to \( y = -2(x - 1)^2 - 3 \). This is the vertex form of a parabola: \( y = a(x - h)^2 + k \).

Step 4: Identify the Vertex and Direction

From the equation \( y = -2(x - 1)^2 - 3 \), identify the vertex \((h, k)\) of the parabola as \((1, -3)\). The negative coefficient \(-2\) indicates the parabola opens downward.

Step 5: Use Graphing Device

Input the equation \( y = -2(x - 1)^2 - 3 \) into your graphing device. Observe the graph's shape and placement: a downward-opening parabola with vertex at \((1, -3)\).

Key Concepts

Vertex Form of a Parabola

The vertex form of a parabola is a way of expressing the quadratic equation that makes identifying the vertex straightforward. In mathematics, this form is written as:

• \( y = a(x - h)^2 + k \)

The values \( h \) and \( k \) represent the coordinates of the vertex of the parabola, \((h, k)\). Moreover, the coefficient \( a \) impact how "wide" or "narrow" the parabola is, as well as its opening direction.

With a positive \( a \), the parabola opens upwards, looking like a smile. If \( a \) is negative, it opens downwards, resembling a frown. This makes identifying not just the vertex's position, but also the general direction and orientation of the parabola quite simple. Try to always rewrite a parabola equation into this form for easier graphing and analysis.

Completing the Square

Completing the square is a vital algebraic technique used to convert a standard quadratic expression into a perfect square trinomial. This process reveals important attributes of the expression and is integral in rewriting the equation into vertex form.

When you have a quadratic like \( ax^2 + bx + c \), the goal is to express it like \( (x + d)^2 + e \). Here’s a general approach to complete the square:

• Ensure the quadratic term's coefficient is 1. If not, factor it out.
• Take the linear coefficient \( b \), divide it by 2, and square the result. This becomes your key transformation number.
• Add and subtract this number within the equation's parentheses to maintain balance.

In our exercise, this helped transform \( x^2 - 2x \) into \( (x - 1)^2 - 1 \), making it easier to see and use in the vertex form. Completing the square not only simplifies graphing but also aids in solving equations and analyzing quadratic relationships.

Graphing Conic Sections

Graphing conic sections involves plotting curves like parabolas, circles, ellipses, and hyperbolas. Each curve has a standard form, which simplifies the plotting process. Graphing these accurately requires an understanding of the specific parameters that define each shape.

Parabolas, a type of conic section, focus on the characteristics derived from their vertex form. Important features include:

• The vertex, which acts like a pinnacle or a valley depending on the direction.
• The axis of symmetry, a line that splits the parabola into mirror images on either side.
• The direction in which the parabola opens, providing an overall visual direction.

Using a graphing device or software by inputting the vertex form equation simplifies observing these features. A precise graph helps in understanding the relationship and trajectory implied by the algebraic equation, allowing for clearer interpretation and analysis.