**Intermediate Algebra**(IA) (

**Algebra 2**) topics below.

Showing posts with label **algebra 2**. Show all posts

Showing posts with label **algebra 2**. Show all posts

This Algebra study guide is designed to supplement your current textbook. Your textbook is well written in a patient style and the appropriate sections should be read before each class meeting.

**Intermediate Algebra** (IA) (**Algebra 2**) topics below.

In this guide, we have solved linear systems using three methods: by graphing, substitution, and elimination. When solving nonlinear systems, we typically choose the substitution method, however, sometimes the other methods work just as well. Remember that to solve a system of equations means to find the common points - if they exist. Given a system of equations consisting of a circle and a line then there can be three possibilities for solutions. Remember that solutions to a system are ordered pairs (x, y). These will be the points where they intersect.

**Solve the nonlinear systems.**

To find the points in common, we will usually use the substitution method. But, as in this case, the graphing and elimination method work just as well.

**Solve**.

**Solve**.

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Given a circle and a parabola there are five possibilities for solutions.

These nonlinear systems given algebraically might look like:To find the points in common, we will usually use the substitution method. But, as in this case, the graphing and elimination method work just as well.

Certainly there are many ways to solve these problems. Experiment with other methods and see if you obtain the same results.

An ellipse is the set of all points whose distances to two fixed points add to the same constant. In this section, we will focus on the equation of ellipses.

The long axis of an ellipse is called the **major axis **and the short axis is called the **minor axis**. These axes terminate at points that we will call vertices. The vertices along the horizontal axis will be (*h *± *a*,*k*) and the vertices along the vertical axes will be (*h*, *k *± *b*). These points, along with the center, will provide us with a method to sketch an ellipse given standard form.

Given the equation of an ellipse in standard form we can graph it using the following steps:

**Graph the ellipse. Label the center and 4 other points.**

**Graph the ellipse. Label the center and four other points.**

**Graph the ellipse. Label the center and four other points.**
**Answer the questions.**

**Example**: Find the equation of the ellipse centered at (3, -1) with a horizontal major axis of length 10 units and vertical minor axis of length 4 units.

**Example**: Find the equation of the ellipse given the vertices ( ±3,0) and (0, ±8).

**Example**: Find the equation of the ellipse whose major axis has vertices (-1, -2) and (-1, 10) and minor axis has co-vertices (-3, 4) and (1, 4).

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[**Interactive**: Ellipses ]

Given the equation of an ellipse in standard form we can graph it using the following steps:

- First plot the center (h, k).
- Plot a point a units up and down from the center.
- Plot a point b units left and right of the center.
- Sketch the ellipse and label the four points.

In this example the major axis is the vertical axis and the minor axis is the horizontal axis. The major axis measures 2*b* = 10 units in length and the minor axis measures 2*a* = 6 units in length. There are no* x*- and *y*- intercepts in this example.

It will often be the case that the ellipse is not given in standard form. In this case, we will begin by rewriting it in standard form.

The next set of examples give the equations in general form. Complete the squares to obtain standard form. Remember to factor the leading coefficient out of each variable grouping before using (B/2)^2 to complete the square.

When the calculation for intercepts yields complex results then there are no real intercepts. This just means that the graph does not cross that particular axis.

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