Graphing Y=x^2+2x-3: A Step-by-Step Solution

Hey guys! Today, we're diving deep into the fascinating world of quadratic equations and their graphical representations. Specifically, we're going to dissect the equation y = x² + 2x - 3 and figure out which graph perfectly portrays its behavior. This is a common type of problem in mathematics, and mastering it will significantly boost your problem-solving skills. So, buckle up, and let's embark on this mathematical journey together!

Understanding Quadratic Equations

Before we jump into the specifics of our equation, let's establish a solid foundation by understanding the basics of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (in our case, 'x') is 2. The general form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The graph of a quadratic equation is always a parabola, a U-shaped curve that opens either upwards or downwards, depending on the sign of the coefficient 'a'. If 'a' is positive, the parabola opens upwards, and if 'a' is negative, it opens downwards. In our equation, y = x² + 2x - 3, 'a' is 1, which is positive, so we know the parabola will open upwards. This is a crucial first step in visualizing the graph. Understanding the general form and the influence of the coefficient 'a' helps us narrow down the possibilities and make educated guesses about the graph's shape and orientation. Moreover, recognizing the symmetry inherent in parabolas is key to pinpointing the vertex and axis of symmetry, which are essential features for accurate graphing.

Analyzing y = x² + 2x - 3

Now, let's focus our attention on the equation at hand: y = x² + 2x - 3. To accurately graph this equation, we need to identify key features that define the parabola. These features include the vertex, the axis of symmetry, and the x and y-intercepts. The vertex is the turning point of the parabola – the point where it changes direction. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. The x-intercepts are the points where the parabola intersects the x-axis (where y = 0), and the y-intercept is the point where the parabola intersects the y-axis (where x = 0). To find the vertex, we can use the formula x = -b / 2a, where 'a' and 'b' are the coefficients from the quadratic equation. In our case, a = 1 and b = 2, so the x-coordinate of the vertex is x = -2 / (2 * 1) = -1. To find the y-coordinate of the vertex, we substitute this x-value back into the equation: y = (-1)² + 2(-1) - 3 = 1 - 2 - 3 = -4. Therefore, the vertex is at the point (-1, -4). The axis of symmetry is the vertical line x = -1. To find the x-intercepts, we set y = 0 and solve for x: 0 = x² + 2x - 3. This quadratic equation can be factored as (x + 3)(x - 1) = 0, which gives us the solutions x = -3 and x = 1. So, the x-intercepts are (-3, 0) and (1, 0). To find the y-intercept, we set x = 0 and solve for y: y = (0)² + 2(0) - 3 = -3. Thus, the y-intercept is (0, -3). With these key features – the vertex, axis of symmetry, and intercepts – we have a comprehensive understanding of the parabola's shape and position on the coordinate plane.

Finding the Vertex

As we just discussed, the vertex is a critical point in determining the graph of a parabola. It represents the minimum or maximum point of the curve. In the equation y = x² + 2x - 3, the coefficient of the x² term is positive (a = 1), indicating that the parabola opens upwards, and the vertex will be the minimum point. To find the vertex, we use the formula x = -b / 2a. Remember, 'a' and 'b' are the coefficients of the quadratic equation in the form ax² + bx + c. In our equation, a = 1 and b = 2. Plugging these values into the formula, we get x = -2 / (2 * 1) = -1. This gives us the x-coordinate of the vertex. To find the y-coordinate, we substitute this x-value back into the original equation: y = (-1)² + 2(-1) - 3 = 1 - 2 - 3 = -4. Therefore, the vertex of the parabola is at the point (-1, -4). This single point provides significant information about the graph. We know the parabola will be symmetrical around the vertical line passing through this point, and it will be the lowest point on the curve. This understanding is incredibly valuable when trying to match the equation to its graph among multiple options. The process of finding the vertex not only helps in plotting the graph accurately but also enhances our comprehension of how the coefficients in the quadratic equation dictate the parabola's position and orientation.

Determining the Intercepts

Another crucial step in graphing the equation y = x² + 2x - 3 is finding the intercepts. Intercepts are the points where the parabola crosses the x-axis and the y-axis. The x-intercepts are the points where y = 0, and the y-intercept is the point where x = 0. To find the x-intercepts, we set y = 0 in the equation and solve for x: 0 = x² + 2x - 3. This is a quadratic equation that can be solved by factoring, completing the square, or using the quadratic formula. In this case, factoring is the simplest method. We need to find two numbers that multiply to -3 and add up to 2. These numbers are 3 and -1. So, we can factor the equation as (x + 3)(x - 1) = 0. Setting each factor equal to zero, we get x + 3 = 0 and x - 1 = 0. Solving these equations gives us x = -3 and x = 1. Therefore, the x-intercepts are (-3, 0) and (1, 0). These points tell us where the parabola crosses the x-axis. To find the y-intercept, we set x = 0 in the original equation: y = (0)² + 2(0) - 3 = -3. So, the y-intercept is (0, -3). This point tells us where the parabola crosses the y-axis. Knowing the intercepts, along with the vertex, gives us a clear picture of the parabola's location and orientation on the coordinate plane. These points act as anchors, guiding us to sketch the curve accurately. Furthermore, understanding how to find intercepts reinforces our ability to manipulate algebraic equations and interpret their graphical representations.

Matching the Graph

Now that we've identified the key features of the equation y = x² + 2x - 3 – the vertex (-1, -4), the x-intercepts (-3, 0) and (1, 0), and the y-intercept (0, -3) – we can confidently match it to the correct graph. Remember, the parabola opens upwards because the coefficient of the x² term is positive. When looking at a set of graphs, we need to find the one that exhibits these characteristics. First, look for a parabola that opens upwards. This eliminates any graphs that open downwards. Next, focus on the vertex. The graph must have its minimum point at (-1, -4). This is a crucial distinguishing feature. Then, verify that the graph intersects the x-axis at -3 and 1, and the y-axis at -3. By systematically checking these features, you can quickly narrow down the options and identify the correct graph. It's like a detective solving a case, each clue (vertex, intercepts, direction) leading you closer to the solution. The ability to connect these algebraic features to their graphical representations is a fundamental skill in mathematics. This exercise not only helps in answering this specific question but also strengthens your overall understanding of quadratic functions and their graphs. So, go ahead, put on your detective hat, and find the graph that perfectly matches our equation!

Conclusion

In conclusion, determining the graph of y = x² + 2x - 3 involves a methodical approach. We began by understanding the basics of quadratic equations and the general shape of a parabola. Then, we focused on identifying key features like the vertex, intercepts, and direction of opening. By using the formula x = -b / 2a to find the vertex, factoring to find the x-intercepts, and direct substitution to find the y-intercept, we gathered all the necessary information to accurately graph the equation. The vertex (-1, -4), x-intercepts (-3, 0) and (1, 0), and y-intercept (0, -3) serve as landmarks that guide us to the correct graph. Matching these features to a visual representation solidifies our understanding of the relationship between algebraic equations and their corresponding graphs. This process is not just about finding the answer; it's about developing a deeper understanding of mathematical concepts and problem-solving strategies. So, the next time you encounter a similar question, remember these steps, and you'll be well-equipped to tackle it with confidence! Keep practicing, keep exploring, and most importantly, keep enjoying the fascinating world of mathematics!