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.