Problem 92 - Precalculus : Building Concepts and Connections

Question

Which of the following points lie(s) on the parabola associated with the function \(f(s)=-s^{2}+6 ?\) Justify your answer. (a) (3,-1) (b) (0,6) (c) (2,1)

Step-by-Step Solution

Answer

Among the given points, only the point (0, 6) lies on the parabola associated with the function \(f(s)=-s^{2}+6\).

Step 1: Validate first point (3, -1)

Take the x-coordinate of the point and substitute it into the equation: \(f(3) = - (3)^2 + 6 = - 9 + 6 = -3\). As it can be seen, the resulting f(s) value does not match the y-coordinate of the point. Therefore, the point (3, -1) does not lie on the parabola.

Step 2: Validate second point (0, 6)

Substitute the x-coordinate of the second point into the equation: \(f(0) = - (0)^2 + 6 = 6\). The resulting f(s) value matches the y-coordinate of the point. Therefore, the point (0, 6) lies on the parabola.

Step 3: Validate third point (2, 1)

Substitute the x-coordinate of third point into the equation: \(f(2) = - (2)^2 + 6 = - 4 + 6 = 2\). The resulting value does not match the y-coordinate of the point. Therefore, the point (2, 1) does not lie on the parabola.

Key Concepts

Substitution Method

The substitution method is a handy tool when dealing with equations to verify if a given point lies on a specific curve, such as a parabola. Here, you replace the variable in the quadratic equation with the x-coordinate of the given point.
By doing so, you calculate the corresponding y-value. This method requires you to plug in the x-value into the quadratic function and simplify it step by step to find the expected y-value.
In our case, the equation is quadratic:

• Given function: \[f(s) = -s^{2} + 6\].
• Substitute the x-value of the point into this equation.
• Simplify to determine if the resulting y matches the point's y-coordinate.

This method helps you easily see which points lie on the parabola.

Coordinate Validation

Once you've used the substitution method to find the theoretical y-values, coordinate validation is the next step.
This involves comparing the y-value calculated by the substitution method with the y-coordinate of the original point.

• If they match, then the point lies on the parabola.
• If they differ, the point does not belong to the curve.

Coordinate validation serves as the conclusive step in verifying the association between a point and a curve.
In our exercise, after substitution:

• For (3, -1), the calculated y does not match -1. Thus, it doesn't fit.
• For (0, 6), the y matches 6. So, this point is on the curve.
• And for (2, 1), the y does not match 1, indicating it isn't on the parabola.

Quadratic Function

A quadratic function is characterized by the presence of an equation in the form: \[f(x) = ax^{2} + bx + c\], where \(a, b, b\) are constants and \(a eq 0\).
In this function, you can visually represent it as a parabola. The shape of the parabola is dictated by the coefficient \(a\):

• If \(a\) is positive, the parabola opens upwards.
• If \(a\) is negative, it opens downwards.

In our example, the function is \[f(s) = -s^{2} + 6\], where \(a = -1\).
Here, the negative sign indicates a downward-opening parabola.
Quadratic functions like this are essential in many applications, from physics to economics, making them crucial for students to understand.