Problem 97 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find three points that lie on the graph of the equation. (There are many correct answers.) $$y=-x^{2}+x-5$$

Step-by-Step Solution

Answer

Three points that lie on the graph of the equation \(y = -x^{2} + x - 5\) are (0, -5), (1, -5), and (2, -7).

Step 1: Choose a Value for x

Start by choosing a value for x. As an example, let's choose \(x=0\). This is the easiest value to choose and will produce a simple y value.

Step 2: Solve for Y

Plug the chosen x value into the equation and solve for y. \(y = -0^{2} + 0 - 5= -5\). Hence, the point (0, -5) lies on the graph of the equation.

Step 3: Repeat Step 1 & 2

Now, repeat step 1 and 2 to find more points. For instance, when \(x = 1\), the corresponding \(y\) will be \(y = -1^{2} + 1 - 5 = -5\). So, (1, -5) is another point that lies on the graph of the equation.

Step 4: Find Third Point

By repeating step 1 and 2 let's find a third point. Choose \(x = 2\). The corresponding \(y\) will be \(y = -2^{2} + 2 - 5 = -7\). Hence, (2, -7) is another point that lies on the graph of the equation.

Key Concepts

Quadratic Functions

Quadratic functions are mathematical expressions of the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are coefficients, and \( a \) is not zero. The graph of a quadratic function creates a parabola that opens upward when \( a > 0 \) and downward when \( a < 0 \).

The general shape of the parabola is symmetrical, and it has a highest or lowest point known as the vertex. The importance of understanding quadratic functions lies in their ability to model various real-world phenomena such as projectile motion or the area of a square.

In our exercise, the quadratic equation \( y = -x^2 + x - 5 \) represents a parabola that opens downward (because the coefficient of \( x^2 \) is negative), and we look for specific points to understand the graph's behavior. The equation is in standard form, making it straightforward to plot on a coordinate grid.

Solving for Y

In the context of graphing equations, 'solving for y' means to find the value of the dependent variable \( y \) for various values of the independent variable \( x \). By doing so, we obtain points that lie on the graph of the equation. It's a crucial step because it allows us to construct the visual representation of the function and better understand its properties.

There are some strategies to make this process more efficient:

• Starting with easy values of \( x \) such as 0 or 1.
• Using symmetry, known for quadratic functions, can provide additional points with less calculation.
• Remembering special points like the vertex, x-intercepts, and y-intercept can serve as key references.

Having solved for \( y \) at \( x = 0, 1, \) and \( 2 \), we've started to reveal the function's shape and can predict its course for further values of \( x \).

Plotting Points

Plotting points on a Cartesian plane is the key action in visualizing the behavior of a function. Once you've solved for \( y \), you can plot the corresponding \( (x, y) \) coordinates on a graph. This step transforms abstract numbers into a concrete visual chart, where the relationship between \( x \) and \( y \) becomes clear.

Consistent plotting leads to the graph's shape emerging. For quadratic functions, after plotting enough points, the characteristic parabola shape is revealed. It's essential to plot points accurately, as even small errors can mislead our interpretation of the function.

To plot the given points effectively, one might:

• Use a scaled grid and carefully mark the \( (x, y) \) coordinates.
• Connect the points smoothly, remembering that a quadratic graph is a curve and not a series of straight lines.
• Verify the plotted points satisfy the original equation, ensuring accuracy.

Through plotting, the solution points (0,-5), (1,-5), and (2,-7) contribute to the visualization of the downward-opening parabola for the equation \( y = -x^2 + x - 5 \).