Solving -x^2+4x=x-4 Graphically: A Step-by-Step Guide

Hey guys! Today, we're diving into a cool method for solving equations: using graphs! Specifically, we'll tackle the equation $-x^2 + 4x = x - 4$. Sonia used this approach, and we're going to break it down so you can understand it too. This method is super useful because it gives you a visual way to see the solutions. Let's get started!

Understanding the Graphical Approach

Before we jump into the specifics, let's understand the core idea. When we have an equation like $-x^2 + 4x = x - 4$, we can think of each side as a separate function. In this case, we have $y = -x^2 + 4x$ and $y = x - 4$. The solutions to the equation are the x-values where the two graphs intersect. Think of it like this: at the points of intersection, the y-values of both functions are equal, which means the equation is satisfied. This graphical method provides a visual representation of the solutions, making it easier to understand and interpret the results. It's not just about crunching numbers; it's about seeing how the functions behave and where they meet. This can be particularly helpful for quadratic equations and other types of equations that might be tricky to solve algebraically. So, keep this key idea in mind: intersections are the solutions! Understanding this foundational concept is crucial for successfully applying the graphical method.

Why Use Graphs to Solve Equations?

So, why bother with graphs when we have algebraic methods? Well, graphs offer several advantages. First, they provide a visual representation of the equation, which can make it easier to understand the behavior of the functions involved. You can see at a glance where the solutions are and how the functions interact. Second, graphical methods can be used to solve equations that are difficult or impossible to solve algebraically. For example, equations involving transcendental functions (like sine, cosine, or exponentials) often require graphical or numerical methods. Third, graphs can help you estimate solutions quickly. Even if you can't find the exact solutions from the graph, you can get a good approximation. This can be especially useful in real-world applications where an approximate solution is sufficient. The graphical approach is all about building a strong conceptual understanding, which is essential for mastering mathematical concepts. By visualizing the problem, you gain a deeper insight into the relationships between variables and functions.

Finally, graphs are incredibly helpful for checking the reasonableness of algebraic solutions. If you've solved an equation algebraically, sketching a quick graph can help you verify that your solutions make sense. If your algebraic solutions don't match the intersections on the graph, it's a sign that you might have made a mistake. For these reasons, mastering the graphical approach is a valuable skill in mathematics. It complements algebraic techniques and provides a more holistic understanding of equation solving.

Step-by-Step Solution

Let's follow Sonia's lead and solve $-x^2 + 4x = x - 4$ graphically. We'll break it down into easy-to-follow steps.

1. Graph the First Equation: $y = -x^2 + 4x$

The first equation, $y = -x^2 + 4x$, is a quadratic equation, which means its graph is a parabola. To graph this, we need to find some key features of the parabola. The most important feature is the vertex, which is the highest or lowest point on the parabola. Since the coefficient of the $x^2$ term is negative (-1), the parabola opens downward, meaning the vertex is the highest point. To find the x-coordinate of the vertex, we can use the formula $x = -b / 2a$, where a and b are the coefficients of the $x^2$ and x terms, respectively. In this case, $a = -1$ and $b = 4$, so the x-coordinate of the vertex is $x = -4 / (2 * -1) = 2$. To find the y-coordinate of the vertex, we substitute $x = 2$ into the equation: $y = -(2)^2 + 4(2) = -4 + 8 = 4$. So, the vertex of the parabola is (2, 4). This point is crucial because it gives us the peak of our curve. The vertex helps us understand the symmetry of the parabola, making it easier to sketch the graph accurately.

Next, we can find the x-intercepts, which are the points where the parabola intersects the x-axis. These are the points where $y = 0$. To find them, we set the equation equal to zero and solve for x: $0 = -x^2 + 4x$. We can factor out an x from the equation: $0 = x(-x + 4)$. This gives us two solutions: $x = 0$ and $-x + 4 = 0$, which means $x = 4$. So, the x-intercepts are (0, 0) and (4, 0). Knowing the intercepts is vital because they anchor the parabola on the x-axis, providing a clear outline of the curve's shape. With the vertex and intercepts, we have a good foundation for sketching the parabola. We can plot these points on a graph and sketch the curve, making sure it's symmetrical around the vertical line that passes through the vertex (the axis of symmetry). This parabola represents all the solutions for the equation $y = -x^2 + 4x$, and it's the first piece of our graphical puzzle. Remember, the more accurately we graph the parabola, the easier it will be to find the intersections with the other equation, leading us to the final solutions. Now, let's move on to the second equation and see how it fits into the picture.

2. Graph the Second Equation: $y = x - 4$

The second equation, $y = x - 4$, is a linear equation, which means its graph is a straight line. Graphing a line is generally easier than graphing a parabola because we only need two points to define a line. One of the simplest ways to graph a line is to find the x- and y-intercepts. The y-intercept is the point where the line intersects the y-axis, which occurs when $x = 0$. Substituting $x = 0$ into the equation, we get $y = 0 - 4 = -4$. So, the y-intercept is (0, -4). This gives us one anchor point on the graph. The y-intercept is particularly useful because it shows where the line crosses the vertical axis, providing a clear starting point for our sketch. To find the x-intercept, we set $y = 0$ and solve for x: $0 = x - 4$, which means $x = 4$. So, the x-intercept is (4, 0). This is the point where the line crosses the horizontal axis, giving us another crucial point.

With the two intercepts, we have enough information to draw the line. We can plot the points (0, -4) and (4, 0) on the same graph as the parabola and then draw a straight line through them. This line represents all the solutions for the equation $y = x - 4$. The line's slope and position give us a clear visual representation of how y changes with respect to x. Alternatively, we could use the slope-intercept form of the equation, $y = mx + b$, where m is the slope and b is the y-intercept. In this case, the slope m is 1, and the y-intercept b is -4. This means the line rises one unit for every one unit it moves to the right. The y-intercept confirms our earlier finding. No matter which method we use, accurately graphing the line is essential for finding the solutions to the original equation. Now that we have both the parabola and the line graphed, we can move on to the most exciting part: finding where they intersect. This intersection will give us the solutions to our equation, so let's take a look and see what we find!

3. Find the Intersection Points

Now for the crucial step: finding the points where the parabola $y = -x^2 + 4x$ and the line $y = x - 4$ intersect. These intersection points are the solutions to the equation $-x^2 + 4x = x - 4$. By looking at the graph, we can visually identify the points where the two curves cross each other. It’s like finding the meeting points of two different paths. If the graph is drawn accurately, we should be able to read the coordinates of these points directly from the graph. Sometimes, the intersections are clear and fall on integer values, making them easy to identify. In other cases, the intersection points might fall between grid lines, and we'll need to estimate their coordinates. Estimation is a valuable skill in graphical solutions, especially when dealing with complex curves.

Let's say, for example, that we observe the graphs intersect at two points: one at $x = -1$ and another at $x = 4$. These x-values are the solutions to our equation. To confirm these solutions, we can substitute them back into both equations to see if the y-values match. For $x = -1$, the parabola gives us $y = -(-1)^2 + 4(-1) = -1 - 4 = -5$, and the line gives us $y = -1 - 4 = -5$. So, the point (-1, -5) is indeed an intersection. For $x = 4$, the parabola gives us $y = -(4)^2 + 4(4) = -16 + 16 = 0$, and the line gives us $y = 4 - 4 = 0$. Thus, the point (4, 0) is also an intersection. These two intersection points confirm that we have found the solutions to the equation. In some cases, the graphs might intersect at only one point, indicating a single solution, or they might not intersect at all, meaning there are no real solutions. The graphical method provides a clear visual check on the number of solutions as well. If you're using graphing software or a calculator, you can often use built-in features to find the intersection points more precisely. These tools can help zoom in on the intersections and provide more accurate coordinates. However, even with technology, it's essential to understand the underlying concept: the intersection points are where the two functions have the same x and y values, and these x-values are the solutions to the equation.

4. State the Solutions

Once we've found the intersection points, the final step is to state the solutions. The solutions to the equation $-x^2 + 4x = x - 4$ are the x-coordinates of the intersection points. Based on our graphical analysis, we found that the graphs intersect at $x = -1$ and $x = 4$. Therefore, the solutions to the equation are $x = -1$ and $x = 4$. To clearly communicate our answer, we should state it explicitly, such as