Quadratic Functions $$ y = x^2 + 2x - 3 $$

Quadratic Functions and Their Graphs

Quadratic functions are an important topic in algebra and are commonly represented by equations in the form of a parabola. In this lesson, we will explore a quadratic function, find its roots, and understand the key features of its graph.

1. Theory of Quadratic Functions

A quadratic function is a polynomial function of degree two. Its general form is y = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a parabola, which can open upward or downward depending on the value of a.

2. Concept Explanation

To understand a quadratic function, we often analyze its roots, vertex, and overall shape. The roots are the x-values where the graph crosses the x-axis, found by setting the function equal to zero. The vertex represents either the lowest or highest point of the parabola.

3. Steps or Formula

1. Write the quadratic function.
2. Set the equation equal to zero to find the roots.
3. Factor the quadratic expression.
4. Solve for the x-values.
5. Identify the vertex and key features of the graph.

4. Example Problem

Given function:

$$ y = x^2 + 2x - 3 $$

Step 1: Find the roots by setting the equation to zero.

$$ x^2 + 2x - 3 = 0 $$

Step 2: Factor the quadratic expression.

$$ (x + 3)(x - 1) = 0 $$

Step 3: Solve for x.

$$ x_1 = -3 \quad \text{and} \quad x_2 = 1 $$

These values are the x-intercepts of the graph.

Step 4: Identify the vertex.

The vertex of the parabola is located at:

$$ (-1, -4) $$

This means the lowest point of the graph occurs when y = -4.

5. Graph Interpretation

The graph of the function y = x² + 2x - 3 is a parabola that opens upward. It crosses the x-axis at x = -3 and x = 1, and has its vertex at the point (-1, -4).

Final Answer:

The quadratic function has roots at x = -3 and x = 1, a vertex at (-1, -4), and its graph is an upward-opening parabola.

Adding Square Roots $$ \sqrt{50} + \sqrt{8} $$

Operations with Radicals: Adding Square Roots

In this lesson, we focus on operations with radicals, specifically adding square roots. Before radicals can be added, each square root must be simplified as much as possible. Only like radicals can be combined.

1. Theory of Adding Square Roots

Square roots can only be added together if they have the same radical part. This means the expression inside the square root must be identical. To achieve this, we often need to simplify each radical first.

2. Concept Explanation

Simplifying a square root involves factoring the number inside the radical into a perfect square multiplied by another number. The square root of the perfect square becomes a whole number, while the remaining factor stays inside the radical.

3. Formula or Steps

1. Simplify each square root separately.
2. Check that the radicals are the same.
3. Add the coefficients.
4. Write the final simplified result.

4. Example Problem

Problem:

$$ \sqrt{50} + \sqrt{8} $$

Solution:

Step 1: Simplify each square root.

$$ \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} $$ $$ \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} $$

Step 2: Add the like radical terms.

$$ 5\sqrt{2} + 2\sqrt{2} = (5 + 2)\sqrt{2} $$ $$ = 7\sqrt{2} $$

Final Answer:

$$ \sqrt{50} + \sqrt{8} = 7\sqrt{2} $$

Operations with Fractions $$ \frac{43}{30} = A + \frac{1}{B + \frac{1}{C + \frac{1}{D}}} $$

Operations with Fractions: Continued Fractions

In this lesson, we explore a more advanced operation with fractions by working with continued fractions. The goal is not only to simplify a fraction, but also to identify specific values hidden within its structure.

1. Problem Overview

We are given the following expression:

$$ \frac{43}{30} = A + \frac{1}{B + \frac{1}{C + \frac{1}{D}}} $$

Our task is to determine the values of A, B, C, and D, then find the value of A + B + C + D.

2. Step-by-Step Solution

Step 1: Convert the fraction into a mixed number.

$$ \frac{43}{30} = 1 + \frac{13}{30} $$

This works because a mixed number can be written as the sum of a whole number and a fraction.

Step 2: Convert the fraction into a continued fraction.

Recall the rule:

$$ \frac{a}{b} = \frac{1}{\frac{b}{a}} $$

Applying this rule to the fraction:

$$ \frac{13}{30} = \frac{1}{\frac{30}{13}} $$

Now simplify the denominator step by step:

$$ \frac{30}{13} = 2 + \frac{4}{13} $$ $$ \frac{4}{13} = \frac{1}{\frac{13}{4}} = \frac{1}{3 + \frac{1}{4}} $$

So the continued fraction becomes:

$$ \frac{13}{30} = \frac{1}{2 + \frac{1}{3 + \frac{1}{4}}} $$

3. Identifying the Values

Substituting back into the original expression:

$$ \frac{43}{30} = 1 + \frac{1}{2 + \frac{1}{3 + \frac{1}{4}}} $$

From this form, we can identify:

• A = 1
• B = 2
• C = 3
• D = 4

4. Final Calculation

Now add all the values:

$$ A + B + C + D = 1 + 2 + 3 + 4 = 10 $$

Final Answer:

$$ A + B + C + D = 10 $$