Finding the Maximum Value of y

Oct 13, 2025 by windailys.com 16 views

When the Given Equation is Graphed in the xy-Plane, for What Value of x Does the Value of y Reach its Maximum?

Understanding the Problem

The problem requires us to find the value of x at which the value of y reaches its maximum when the given equation is graphed in the xy-plane. To solve this problem, we need to first complete the given equation and then analyze its graph to determine the maximum value of y.

Completing the Equation

The given equation is incomplete, but based on the context, it appears to be a linear equation in the form of y = mx + b, where m is the slope and b is the y-intercept. Assuming the equation is y = -x + 6x, we can simplify it to y = 5x.

Analyzing the Graph

To analyze the graph of the equation y = 5x, we need to understand the characteristics of a linear equation. A linear equation has a straight line graph, and the slope of the line represents the rate of change of y with respect to x.

Finding the Maximum Value of y

Since the equation y = 5x is a linear equation, its graph is a straight line with a positive slope. This means that as x increases, y also increases. Therefore, there is no maximum value of y, and the value of y will continue to increase as x increases.

However, if we consider a quadratic equation

If we consider a quadratic equation in the form of y = ax^2 + bx + c, where a, b, and c are constants, the graph of the equation will be a parabola. The maximum value of y will occur at the vertex of the parabola.

Finding the Vertex of the Parabola

To find the vertex of the parabola, we need to use the formula x = -b / 2a. This formula gives us the x-coordinate of the vertex.

Example: Finding the Maximum Value of y

Let's consider the quadratic equation y = x^2 - 4x + 3. To find the maximum value of y, we need to find the vertex of the parabola.

Step 1: Identify the values of a, b, and c

In the equation y = x^2 - 4x + 3, a = 1, b = -4, and c = 3.

Step 2: Use the formula to find the x-coordinate of the vertex

x = -b / 2a x = -(-4) / 2(1) x = 4 / 2 x = 2

Step 3: Find the y-coordinate of the vertex

To find the y-coordinate of the vertex, we need to substitute the x-coordinate into the equation.

y = x^2 - 4x + 3 y = (2)^2 - 4(2) + 3 y = 4 - 8 + 3 y = -1

Conclusion

In conclusion, the value of x at which the value of y reaches its maximum depends on the type of equation. For a linear equation, there is no maximum value of y, and the value of y will continue to increase as x increases. For a quadratic equation, the maximum value of y occurs at the vertex of the parabola, and we can find the x-coordinate of the vertex using the formula x = -b / 2a.

Understanding the Graph of a Quadratic Equation

A quadratic equation in the form of y = ax^2 + bx + c has a graph that is a parabola. The parabola can open upwards or downwards, depending on the value of a.

Upward-Opening Parabola

If a > 0, the parabola opens upwards, and the vertex is the minimum point of the parabola.

Downward-Opening Parabola

If a < 0, the parabola opens downwards, and the vertex is the maximum point of the parabola.

Finding the Vertex of a Parabola

To find the vertex of a parabola, we need to use the formula x = -b / 2a. This formula gives us the x-coordinate of the vertex.

Example: Finding the Vertex of a Parabola

Let's consider the quadratic equation y = -x^2 + 4x - 3. To find the vertex of the parabola, we need to use the formula x = -b / 2a.

Step 1: Identify the values of a, b, and c

In the equation y = -x^2 + 4x - 3, a = -1, b = 4, and c = -3.

Step 2: Use the formula to find the x-coordinate of the vertex

x = -b / 2a x = -4 / 2(-1) x = -4 / -2 x = 2

Step 3: Find the y-coordinate of the vertex

To find the y-coordinate of the vertex, we need to substitute the x-coordinate into the equation.

y = -x^2 + 4x - 3 y = -(2)^2 + 4(2) - 3 y = -4 + 8 - 3 y = 1

Conclusion

Graphing the Equation

To graph the equation y = -x^2 + 4x - 3, we need to plot the points on the graph and draw a smooth curve through the points.

Step 1: Plot the points

To plot the points, we need to substitute different values of x into the equation and find the corresponding values of y.

Step 2: Draw a smooth curve

Once we have plotted the points, we need to draw a smooth curve through the points to represent the graph of the equation.

Graphing the Equation

The graph of the equation y = -x^2 + 4x - 3 is a downward-opening parabola. The vertex of the parabola is at the point (2, 1).

Conclusion

Understanding the Graph of a Linear Equation

A linear equation in the form of y = mx + b has a graph that is a straight line. The slope of the line represents the rate of change of y with respect to x.

Finding the Slope of a Line

To find the slope of a line, we need to use the formula m = (y2 - y1) / (x2 - x1). This formula gives us the slope of the line.

Example: Finding the Slope of a Line

Let's consider the linear equation y = 2x + 3. To find the slope of the line, we need to use the formula m = (y2 - y1) / (x2 - x1).

Step 1: Identify the points

In the equation y = 2x + 3, we can choose any two points on the line. Let's choose the points (0, 3) and (1, 5).

Step 2: Use the formula to find the slope

m = (y2 - y1) / (x2 - x1) m = (5 - 3) / (1 - 0) m = 2 / 1 m = 2

Conclusion

Understanding the Graph of a Quadratic Equation

A quadratic equation in the form of y = ax^2 + bx + c has a graph that is a parabola. The parabola can open upwards or downwards, depending on the value of a.

Upward-Opening Parabola

If a > 0, the parabola opens upwards, and the vertex is the minimum point of the parabola.

Downward-Opening Parabola

If a < 0, the parabola opens downwards, and the vertex is the maximum point of the parabola.

Finding the Vertex of a Parabola

To find the vertex of a parabola, we need to use the formula x = -b / 2a. This formula gives us the x-coordinate of the vertex.

Example: Finding the Vertex of a Parabola

Let's consider the quadratic equation y = -x^2 + 4x - 3. To find the vertex of the parabola, we need to use the formula x = -b /
Q&A: When the Given Equation is Graphed in the xy-Plane, for What Value of x Does the Value of y Reach its Maximum?

Q: What is the main difference between a linear equation and a quadratic equation?

A: The main difference between a linear equation and a quadratic equation is the shape of their graphs. A linear equation has a straight line graph, while a quadratic equation has a parabola graph.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point at which the parabola changes direction. It is the maximum or minimum point of the parabola, depending on whether the parabola opens upwards or downwards.

Q: How do we find the vertex of a parabola?

A: To find the vertex of a parabola, we need to use the formula x = -b / 2a. This formula gives us the x-coordinate of the vertex.

Q: What is the significance of the slope of a line?

A: The slope of a line represents the rate of change of y with respect to x. It tells us how steep the line is and whether it is increasing or decreasing.

Q: How do we find the slope of a line?

A: To find the slope of a line, we need to use the formula m = (y2 - y1) / (x2 - x1). This formula gives us the slope of the line.

Q: What is the difference between a maximum and a minimum point?

A: A maximum point is the highest point on a graph, while a minimum point is the lowest point on a graph.

Q: How do we determine whether a parabola opens upwards or downwards?

A: We can determine whether a parabola opens upwards or downwards by looking at the value of a in the equation y = ax^2 + bx + c. If a > 0, the parabola opens upwards, and if a < 0, the parabola opens downwards.

Q: What is the significance of the y-intercept of a line?

A: The y-intercept of a line is the point at which the line intersects the y-axis. It tells us the value of y when x is equal to 0.

Q: How do we find the y-intercept of a line?

A: To find the y-intercept of a line, we need to look at the equation of the line and find the value of b. The y-intercept is equal to the value of b.

Q: What is the difference between a linear equation and a quadratic equation in terms of their graphs?

A: A linear equation has a straight line graph, while a quadratic equation has a parabola graph. A linear equation has a constant rate of change, while a quadratic equation has a changing rate of change.

Q: How do we determine whether a line is increasing or decreasing?

A: We can determine whether a line is increasing or decreasing by looking at the slope of the line. If the slope is positive, the line is increasing, and if the slope is negative, the line is decreasing.

Q: What is the significance of the vertex of a parabola in terms of optimization problems?

A: The vertex of a parabola is the maximum or minimum point of the parabola, and it is often used in optimization problems to find the maximum or minimum value of a function.

Q: How do we use the vertex of a parabola to solve optimization problems?

A: We can use the vertex of a parabola to solve optimization problems by finding the x-coordinate of the vertex and substituting it into the equation to find the corresponding value of y.

Q: What is the difference between a maximum and a minimum point in terms of optimization problems?

A: A maximum point is the highest point on a graph, while a minimum point is the lowest point on a graph. In optimization problems, we often want to find the maximum or minimum value of a function, and the vertex of a parabola can help us do this.

Q: How do we determine whether a parabola is opening upwards or downwards in terms of optimization problems?

A: We can determine whether a parabola is opening upwards or downwards by looking at the value of a in the equation y = ax^2 + bx + c. If a > 0, the parabola opens upwards, and if a < 0, the parabola opens downwards.

Q: What is the significance of the slope of a line in terms of optimization problems?

A: The slope of a line represents the rate of change of y with respect to x, and it can be used to determine whether a line is increasing or decreasing.

Q: How do we use the slope of a line to solve optimization problems?

A: We can use the slope of a line to solve optimization problems by finding the slope of the line and using it to determine whether the line is increasing or decreasing.

Q: What is the difference between a linear equation and a quadratic equation in terms of their applications?

A: A linear equation is often used to model linear relationships between variables, while a quadratic equation is often used to model quadratic relationships between variables.

Q: How do we determine whether a linear equation or a quadratic equation is more suitable for a given problem?

A: We can determine whether a linear equation or a quadratic equation is more suitable for a given problem by looking at the nature of the relationship between the variables. If the relationship is linear, a linear equation is more suitable, and if the relationship is quadratic, a quadratic equation is more suitable.