Graph Intersection How Many Times Does Y=-2x^2+3x+5 Intersect The X-axis
igma Let's explore the question of how many times the graph of the function y = -2x² + 3x + 5 intersects or touches the x-axis. This is a fundamental concept in algebra, and understanding how to determine the points of intersection with the x-axis is crucial for analyzing quadratic functions and their graphical representations. In this article, we will delve into the methods used to solve this problem, providing a comprehensive explanation and a step-by-step approach to arrive at the correct answer.
Understanding the X-Axis Intersections
In mathematics, the x-axis intersections, also known as the roots or zeros of a function, are the points where the graph of the function crosses or touches the x-axis. At these points, the value of y is zero. For a quadratic function in the form of y = ax² + bx + c, the x-axis intersections can be found by setting y to zero and solving the resulting quadratic equation:
ax² + bx + c = 0
The solutions to this equation represent the x-coordinates where the graph intersects the x-axis. The number of real solutions determines how many times the graph intersects or touches the x-axis. A quadratic equation can have zero, one, or two real solutions, which correspond to the graph not intersecting the x-axis, touching it at one point (tangent), or intersecting it at two points, respectively.
Methods to Determine the Number of Intersections
There are several methods to determine the number of times a quadratic function intersects the x-axis. The most common methods include:
- Solving the Quadratic Equation: We can solve the quadratic equation ax² + bx + c = 0 using factoring, completing the square, or the quadratic formula. The solutions will give us the x-coordinates of the intersection points. If we find two distinct real solutions, the graph intersects the x-axis twice. If we find one real solution (a repeated root), the graph touches the x-axis at one point. If we find no real solutions (complex roots), the graph does not intersect the x-axis.
- Using the Discriminant: The discriminant is a part of the quadratic formula that provides valuable information about the nature of the roots without actually solving the equation. The discriminant (Δ) is given by the formula:
Δ = b² - 4ac
The discriminant helps us determine the number of real solutions:
* If Δ > 0, the equation has two distinct real solutions, and the graph intersects the x-axis twice.
* If Δ = 0, the equation has one real solution (a repeated root), and the graph touches the x-axis at one point.
* If Δ < 0, the equation has no real solutions, and the graph does not intersect the x-axis.
- Graphical Method: We can graph the quadratic function and visually inspect the graph to see how many times it intersects the x-axis. This method provides a visual confirmation of the algebraic results.
Applying the Methods to the Given Function: y = -2x² + 3x + 5
Now, let's apply these methods to the given function, y = -2x² + 3x + 5, to determine how many times its graph intersects the x-axis.
1. Using the Discriminant
The function is given by y = -2x² + 3x + 5. Here, a = -2, b = 3, and c = 5. We will use the discriminant formula to find Δ:
Δ = b² - 4ac
Substitute the values:
Δ = (3)² - 4(-2)(5)
Δ = 9 + 40
Δ = 49
Since Δ = 49, which is greater than 0, the quadratic equation has two distinct real solutions. This means the graph of the function intersects the x-axis twice.
2. Solving the Quadratic Equation
To confirm our result, let's solve the quadratic equation -2x² + 3x + 5 = 0 using the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
Substitute the values a = -2, b = 3, and c = 5:
x = (-3 ± √(3² - 4(-2)(5))) / (2(-2))
x = (-3 ± √(9 + 40)) / (-4)
x = (-3 ± √49) / (-4)
x = (-3 ± 7) / (-4)
We have two possible solutions:
- x = (-3 + 7) / (-4) = 4 / (-4) = -1
- x = (-3 - 7) / (-4) = -10 / (-4) = 2.5
Thus, the solutions are x = -1 and x = 2.5. Since we have two distinct real solutions, the graph intersects the x-axis at two points.
3. Graphical Method
To visualize this, we can sketch the graph of y = -2x² + 3x + 5. Since a = -2 is negative, the parabola opens downward. The x-intercepts are at x = -1 and x = 2.5, which confirms that the graph intersects the x-axis twice. The vertex of the parabola can be found using the formula x_vertex = -b / (2a):
x_vertex = -3 / (2(-2)) = 3 / 4 = 0.75
The y-coordinate of the vertex can be found by substituting x_vertex into the equation:
y_vertex = -2(0.75)² + 3(0.75) + 5
y_vertex = -2(0.5625) + 2.25 + 5
y_vertex = -1.125 + 2.25 + 5
y_vertex = 6.125
So, the vertex is at (0.75, 6.125). This information helps us sketch a more accurate graph, further confirming that the parabola intersects the x-axis at two points.
Conclusion
By using the discriminant and solving the quadratic equation, we have determined that the graph of the function y = -2x² + 3x + 5 intersects the x-axis at two points. The discriminant (Δ = 49) being greater than 0 indicates two real solutions, and solving the quadratic equation confirms these solutions as x = -1 and x = 2.5. The graphical method also supports this conclusion by visually showing the parabola intersecting the x-axis twice.
Therefore, the correct answer is D. 2.
Understanding these methods and applying them to quadratic functions is essential for solving various problems in algebra and calculus. Whether you use the discriminant, solve the equation, or sketch the graph, each approach provides valuable insights into the behavior of the function and its intersections with the x-axis.
