X-Intercept, Y-Intercept, And Turning Point Of X² - 2x - 3 = 0
Alright guys, let's break down how to find the x-intercept, y-intercept, and turning point (also known as the vertex) of the quadratic equation x² - 2x - 3 = 0. This is a classic problem in algebra, and understanding these concepts is super useful for graphing and analyzing quadratic functions. So, let's dive right in!
Finding the X-Intercept(s)
X-intercepts, also known as roots or zeros, are the points where the parabola intersects the x-axis. At these points, the value of y is zero. To find the x-intercepts, we need to solve the quadratic equation x² - 2x - 3 = 0. There are a few ways to do this, but factoring is often the easiest method when it's possible.
Factoring the Quadratic Equation
The goal of factoring is to rewrite the quadratic equation in the form (x - a)(x - b) = 0, where 'a' and 'b' are the roots of the equation. Let's factor x² - 2x - 3 = 0. We're looking for two numbers that multiply to -3 and add up to -2. Those numbers are -3 and 1. Therefore, we can rewrite the equation as:
(x - 3)(x + 1) = 0
Solving for x
Now that we have the factored form, we can easily find the x-intercepts. For the product of two factors to be zero, at least one of the factors must be zero. So, we set each factor equal to zero and solve for x:
x - 3 = 0 => x = 3 x + 1 = 0 => x = -1
Therefore, the x-intercepts are x = 3 and x = -1. These correspond to the points (3, 0) and (-1, 0) on the graph.
Using the Quadratic Formula (If Factoring Isn't Obvious)
If you can't easily factor the quadratic equation, don't worry! The quadratic formula is your best friend. For a quadratic equation in the form ax² + bx + c = 0, the quadratic formula is:
x = (-b ± √(b² - 4ac)) / (2a)
In our case, a = 1, b = -2, and c = -3. Plugging these values into the formula, we get:
x = (2 ± √((-2)² - 4 * 1 * -3)) / (2 * 1) x = (2 ± √(4 + 12)) / 2 x = (2 ± √16) / 2 x = (2 ± 4) / 2
So, we have two possible solutions:
x = (2 + 4) / 2 = 6 / 2 = 3 x = (2 - 4) / 2 = -2 / 2 = -1
Again, we find that the x-intercepts are x = 3 and x = -1, confirming our earlier result from factoring. Knowing how to find x-intercepts is crucial for understanding the behavior of the quadratic function and its graph. Remember, the x-intercepts are the points where the parabola crosses the x-axis, and they represent the solutions to the quadratic equation.
Finding the Y-Intercept
The y-intercept is the point where the parabola intersects the y-axis. At this point, the value of x is zero. To find the y-intercept, we simply substitute x = 0 into the quadratic equation.
Substituting x = 0
Let's substitute x = 0 into the equation x² - 2x - 3 = 0:
y = (0)² - 2(0) - 3 y = 0 - 0 - 3 y = -3
Therefore, the y-intercept is y = -3. This corresponds to the point (0, -3) on the graph.
Understanding the Y-Intercept
The y-intercept is particularly easy to find because it's simply the constant term in the quadratic equation when the equation is in the standard form (ax² + bx + c = 0). In this case, the constant term is -3, so the y-intercept is (0, -3). Finding the y-intercept provides another key point for graphing the parabola and understanding its position relative to the coordinate axes. The y-intercept is the point where the parabola crosses the y-axis, and it gives us a direct indication of the vertical shift of the graph.
Finding the Turning Point (Vertex)
The turning point, also known as the vertex, is the point where the parabola changes direction. It's either the minimum point (if the parabola opens upwards) or the maximum point (if the parabola opens downwards). To find the turning point, we need to find the x-coordinate of the vertex first, and then substitute that value back into the equation to find the y-coordinate.
Finding the X-Coordinate of the Vertex
The x-coordinate of the vertex can be found using the formula:
x_vertex = -b / (2a)
In our case, a = 1 and b = -2. Plugging these values into the formula, we get:
x_vertex = -(-2) / (2 * 1) x_vertex = 2 / 2 x_vertex = 1
So, the x-coordinate of the vertex is x = 1.
Finding the Y-Coordinate of the Vertex
Now that we have the x-coordinate of the vertex, we can substitute it back into the quadratic equation to find the y-coordinate:
y_vertex = (1)² - 2(1) - 3 y_vertex = 1 - 2 - 3 y_vertex = -4
Therefore, the y-coordinate of the vertex is y = -4. This means the turning point (vertex) is at the point (1, -4).
Understanding the Turning Point
The turning point (1, -4) is the minimum point of the parabola because the coefficient of the x² term (a) is positive (a = 1). This means the parabola opens upwards. The turning point is a critical feature of the parabola because it represents the lowest (or highest) point on the graph. Knowing the turning point helps us understand the range of the quadratic function and its overall shape. The turning point is also the axis of symmetry for the parabola, meaning that the graph is symmetrical around the vertical line x = 1.
Summary
To recap, for the quadratic equation x² - 2x - 3 = 0, we found the following:
- X-intercepts: (3, 0) and (-1, 0)
- Y-intercept: (0, -3)
- Turning Point (Vertex): (1, -4)
Understanding how to find these key points is essential for graphing and analyzing quadratic functions. Whether you use factoring, the quadratic formula, or the vertex formula, these techniques will help you gain a deeper understanding of quadratic equations and their corresponding parabolas. Keep practicing, and you'll become a pro in no time! Remember to always double-check your work and use different methods to verify your results. Happy solving!