Graphing The Solution Of Y ≤ -x^2 + 2x A Comprehensive Guide

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To effectively graph the solution of the inequality ${ y \le -x^2 + 2x }$, we need to follow a structured approach. This involves understanding the nature of the quadratic function, finding key points, and interpreting the inequality. This comprehensive guide will walk you through each step, ensuring you grasp the underlying concepts and can confidently apply them to similar problems. The primary goal here is to visualize all the points ${ (x, y) }$ that satisfy the given inequality. This process not only enhances your algebraic skills but also significantly improves your ability to interpret graphical representations of mathematical relationships.

1. Understanding the Quadratic Function

Identifying the Parabola

At the heart of the inequality ${ y \le -x^2 + 2x }$ lies a quadratic function. Understanding quadratic functions is crucial for graphing inequalities of this nature. The function ${ y = -x^2 + 2x }$ represents a parabola. The general form of a quadratic function is ${ y = ax^2 + bx + c }$, where ${ a }$, ${ b }$, and ${ c }$ are constants. In our case, ${ a = -1 }$, ${ b = 2 }$, and ${ c = 0 }$. The coefficient ${ a }$ plays a vital role in determining the parabola’s orientation. When ${ a < 0 }$, as in our example, the parabola opens downward, indicating that it has a maximum point. Conversely, if ${ a > 0 }$, the parabola opens upward, possessing a minimum point. This initial observation sets the stage for further analysis.

Key Features of the Parabola

To accurately graph the parabola, we need to identify its key features, namely the vertex, the axis of symmetry, and the x-intercepts (if they exist). These elements serve as the foundational points for sketching the curve. The vertex is the highest or lowest point on the parabola and is a critical reference point. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. This symmetry simplifies the graphing process, as points on one side of the axis mirror those on the other side. Lastly, the x-intercepts are the points where the parabola intersects the x-axis, providing additional anchor points for the graph.

2. Finding the Vertex

Using the Vertex Formula

The vertex of a parabola given by ${ y = ax^2 + bx + c }$ can be found using the vertex formula. This formula provides the x-coordinate of the vertex, which can then be substituted back into the equation to find the corresponding y-coordinate. The x-coordinate of the vertex, denoted as ${ h }$, is given by ${ h = -b / (2a) }$. For our function, ${ y = -x^2 + 2x }$, we have ${ a = -1 }$ and ${ b = 2 }$. Plugging these values into the formula, we get:

${ h = -2 / (2 * -1) = 1 }$

This tells us that the x-coordinate of the vertex is 1. Now, to find the y-coordinate, denoted as ${ k }$, we substitute ${ x = 1 }$ back into the original equation:

${ k = -(1)^2 + 2(1) = -1 + 2 = 1 }$

Therefore, the vertex of the parabola is at the point ${ (1, 1) }$. This point is the maximum value of the function, as the parabola opens downward. Accurately determining the vertex is pivotal in graphing the inequality, as it serves as the central reference point around which the rest of the parabola is sketched.

Significance of the Vertex

The vertex not only provides the maximum or minimum value of the quadratic function but also helps in determining the range of the function. Since our parabola opens downward and has a vertex at ${ (1, 1) }$, the maximum value of ${ y }$ is 1. This understanding is crucial when interpreting the inequality ${ y \le -x^2 + 2x }$, as it tells us that we are interested in the region below or on the parabola. The vertex serves as a critical point for defining the boundary of the solution region.

3. Determining the Axis of Symmetry

Understanding Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two mirror-image halves. Identifying the axis of symmetry simplifies the graphing process because once you plot points on one side of the axis, you can easily mirror them onto the other side. The equation of the axis of symmetry is given by ${ x = h }$, where ${ h }$ is the x-coordinate of the vertex.

Calculating the Axis of Symmetry

In our case, the x-coordinate of the vertex is 1, as we calculated earlier. Therefore, the equation of the axis of symmetry is ${ x = 1 }$. This vertical line acts as a guide for plotting points symmetrically around the vertex. For instance, if we find a point at ${ (0, 0) }$ on the parabola, we can immediately plot another point at ${ (2, 0) }$ since it is equidistant from the axis of symmetry on the opposite side. The axis of symmetry significantly reduces the number of calculations needed to sketch the parabola accurately.

Role in Graphing

The axis of symmetry provides a line of reflection, making the graphing process more efficient. When plotting points, choosing x-values on either side of the axis of symmetry allows you to quickly generate corresponding y-values. This symmetry ensures that your sketch accurately represents the parabolic shape. Moreover, the axis of symmetry helps in visualizing the parabola’s overall structure and behavior, further aiding in the interpretation of the inequality.

4. Finding the X-Intercepts

Setting ${ y = 0 }$

The x-intercepts are the points where the parabola intersects the x-axis. Determining the x-intercepts is a crucial step in graphing the parabola, as they provide additional anchor points for the curve. To find the x-intercepts, we set ${ y = 0 }$ in the equation ${ y = -x^2 + 2x }$ and solve for ${ x }$. This gives us the equation:

${ 0 = -x^2 + 2x }$

Solving the Quadratic Equation

We can solve this quadratic equation by factoring. Factoring the equation ${ -x^2 + 2x = 0 }$, we get:

${ x(-x + 2) = 0 }$

This equation is satisfied when either ${ x = 0 }$ or ${ -x + 2 = 0 }$. Solving for ${ x }$ in the second case gives us:

${ -x = -2 }$

${ x = 2 }$

Thus, the x-intercepts are ${ x = 0 }$ and ${ x = 2 }$. These points, ${ (0, 0) }$ and ${ (2, 0) }$, are where the parabola crosses the x-axis. The x-intercepts provide essential information about the parabola’s position and shape.

Importance of X-Intercepts

The x-intercepts, along with the vertex, define the parabola’s key characteristics. Knowing where the parabola intersects the x-axis helps in sketching the curve accurately. Additionally, the x-intercepts can be used to determine the domain intervals where the parabola is above or below the x-axis, which is particularly useful when dealing with inequalities. In the context of the inequality ${ y \le -x^2 + 2x }$, the x-intercepts help delineate the region where the y-values are less than or equal to the quadratic function.

5. Plotting the Parabola

Utilizing Key Points

With the vertex at ${ (1, 1) }$ and the x-intercepts at ${ (0, 0) }$ and ${ (2, 0) }$, we have sufficient information to sketch the parabola. Plotting these key points on a coordinate plane is the first step in visualizing the curve. The vertex serves as the maximum point, and the x-intercepts define where the parabola intersects the x-axis. These points act as anchors, guiding the shape of the parabola.

Sketching the Curve

Draw a smooth curve that passes through the plotted points. Remember that the parabola is symmetrical about the line ${ x = 1 }$, so the shape on one side of the axis of symmetry mirrors the shape on the other side. Since the coefficient of ${ x^2 }$ is negative, the parabola opens downward. Sketching the curve involves connecting the points in a parabolic shape, ensuring it is smooth and symmetrical. The more accurately the key points are plotted, the more accurate the sketch of the parabola will be.

Importance of Accuracy

Accuracy in plotting the parabola is crucial, especially when dealing with inequalities. The parabola acts as the boundary for the solution region, and any inaccuracies in the sketch can lead to an incorrect representation of the solution set. Pay close attention to the vertex, x-intercepts, and the overall shape of the parabola to ensure the graph is as precise as possible. An accurate parabola is essential for correctly interpreting and graphing the inequality.

6. Interpreting the Inequality

Understanding ${ y \le }$

The inequality ${ y \le -x^2 + 2x }$ means that we are looking for all points ${ (x, y) }$ where the y-coordinate is less than or equal to the value of the quadratic function at that x-coordinate. Interpreting the inequality involves understanding that ${ y \le }$ indicates the region below or on the parabola. This is because for any given x-value, we want all the y-values that are less than or equal to the corresponding y-value on the parabola.

Shading the Solution Region

To represent the solution region, we shade the area below the parabola. This shaded region includes all points ${ (x, y) }$ that satisfy the inequality. Since the inequality includes “equal to,” the parabola itself is also part of the solution. Therefore, we draw a solid line for the parabola to indicate that points on the curve are included in the solution set. Shading the solution region visually represents the infinite set of points that satisfy the inequality.

Solid vs. Dashed Line

It is important to note that if the inequality were ${ y < -x^2 + 2x }$, we would use a dashed line to represent the parabola. A dashed line indicates that the points on the parabola are not included in the solution set. The choice between a solid and dashed line is crucial for accurately representing the solution to the inequality. Using the correct line type ensures that the graph precisely communicates which points are included in the solution.

7. Shading the Correct Region

Testing Points

To ensure we have shaded the correct region, we can test a point that is not on the parabola. A simple point to test is the origin ${ (0, 0) }$. Substituting ${ x = 0 }$ and ${ y = 0 }$ into the inequality ${ y \le -x^2 + 2x }$, we get:

${ 0 \le -(0)^2 + 2(0) }$

${ 0 \le 0 }$

This statement is true, which means that the origin is part of the solution region. Testing points is a reliable method for verifying that the correct area has been shaded. If the test point satisfies the inequality, then the region containing that point should be shaded. If it does not satisfy the inequality, then the opposite region should be shaded.

Identifying the Correct Area

Since the origin is below the vertex and satisfies the inequality, we shade the region below the parabola. This shaded area, along with the solid parabola line, represents the complete solution set for ${ y \le -x^2 + 2x }$. Identifying the correct area involves understanding the inequality and using test points to confirm the solution region. The shaded area visually communicates all the points that meet the condition specified by the inequality.

Final Check

As a final check, consider the inequality ${ y \le -x^2 + 2x }$. The parabola opens downwards, and the inequality indicates that we are interested in the region below or on the parabola. The vertex is at ${ (1, 1) }$, and the x-intercepts are at ${ (0, 0) }$ and ${ (2, 0) }$. We have drawn a solid parabola to indicate inclusion and shaded the area below the parabola. This final check ensures that all aspects of the solution have been correctly represented.

Conclusion

Graphing the solution to the inequality ${ y \le -x^2 + 2x }$ involves several key steps: understanding the quadratic function, finding the vertex and x-intercepts, plotting the parabola, interpreting the inequality, and shading the correct region. Each step is crucial for accurately representing the solution set. By following this comprehensive guide, you can confidently graph quadratic inequalities and interpret their solutions. Mastering these steps enhances your understanding of algebraic concepts and improves your ability to visualize mathematical relationships. The combination of analytical techniques and graphical representation provides a powerful tool for solving mathematical problems and gaining deeper insights into the behavior of functions and inequalities.