Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. $$ \text { Graph } x^{2}-4 x+y^{2}+10 y-7=0 $$

Short answer

The graph is a circle with center (2, -5) and radius 6.

Step-by-step solution

- Identify and rearrange the equation

First, rewrite the given quadratic equation in the standard form to identify the conic section more easily: \[ x^2 - 4x + y^2 + 10y - 7 = 0 \] Our goal is to complete the square for both \(x\) and \(y\) terms to transform the equation into a standard circle or ellipse form.

- Complete the square for x

Focus on the \(x\) terms: \[ x^2 - 4x \] Add and subtract the square of half the coefficient of \(x\): \[ x^2 - 4x + 4 - 4 \] This can be written as: \[ (x - 2)^2 - 4 \]

- Complete the square for y

Next, focus on the \(y\) terms: \[ y^2 + 10y \] Add and subtract the square of half the coefficient of \(y\): \[ y^2 + 10y + 25 - 25 \] This can be written as: \[ (y + 5)^2 - 25 \]

- Combine the completed squares

Now combine the completed squares and simplify the equation: \[ (x - 2)^2 - 4 + (y + 5)^2 - 25 - 7 = 0 \] Combine and simplify the constants: \[ (x - 2)^2 + (y + 5)^2 - 36 = 0 \] \[ (x - 2)^2 + (y + 5)^2 = 36 \]

- Identify the conic section

The equation is now in the standard form of a circle: \[ (x - 2)^2 + (y + 5)^2 = 6^2 \] This represents a circle with center \((2, -5)\) and radius \(6\).

- Graph the circle

Plot the center of the circle at \((2, -5)\). From this center, draw a circle with radius \(6\), which means extending 6 units in all directions (up, down, left, and right) from the center.

Key concepts

Quadratic Equations

Quadratic equations are polynomial equations of degree 2. They have the general form: ax^2 + bx + c = 0 where a, b, and c are constants with a ≠ 0. Quadratic equations form parabolas when graphed on the xy-plane. Many problems involving quadratic equations can be simplified using completing the square. This technique helps convert the general form into a format that provides more insight into the properties of the equation, such as vertex, axis of symmetry, and more.

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane. There are four main types of conic sections: circles, ellipses, parabolas, and hyperbolas. Each of these can be represented by specific quadratic equations. Understanding conic sections is essential as they appear in various algebra and geometry problems. Let's explore the circle, one of the simplest conic sections.

Circle Equation

The general equation of a circle is: (x - h)^2 + (y - k)^2 = r^2 where (h, k) represents the center of the circle, and r denotes its radius. If the equation of a circle is not in this standard form, you can complete the square to rewrite it as such. For example, the given problem: x^2 - 4x + y^2 + 10y - 7 = 0 can be transformed by completing the square into: (x - 2)^2 + (y + 5)^2 = 6^2 From this form, it's clear that the circle has a center at (2, -5) and a radius of 6.

Graphing

Graphing involves plotting equations on a coordinate grid to visualize solutions and better understand the geometric properties of functions. To graph the equation of a circle:

• Identify the center and radius from the standard form of the circle equation.
• Mark the center on a graph.
• From the center, measure and draw points that are the radius distance away in all directions.
• Connect these points smoothly to form a circle.

In our example, the circle with center at (2, -5) and radius 6 can be drawn by marking (2, -5) and extending 6 units in all directions, ensuring a smooth, round shape.