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Chapter 2: Problem 92

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)

Short Answer

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

Step by step solution

01

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.
02

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.
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

In Exercises \(101-104,\) let \(f(t)=3 t+1\) and \(g(x)=x^{2}+4\). Evaluate \((g \circ g)\left(\frac{1}{2}\right)\)

In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=x-2 ; g(x)=2 x^{2}-x+3$$

Is it possible for a quadratic function to have the set of all real numbers as its range? Explain. (Hint: Examine the graph of a general quadratic function.)

In Exercises \(97-100,\) let \(f(t)=-t^{2}\) and \(g(x)=x^{2}-1\). Find an expression for \((g \circ g)(x),\) and give the domain of \(g \circ g\).

The shape of the Gateway Arch in St. Louis, Missouri, can be approximated by a parabola. (The actual shape will be discussed in an exercise in the chapter on exponential functions.) The highest point of the arch is approximately 625 feet above the ground, and the arch has a (horizontal) span of approximately 600 feet. (Check your book to see image) (a) Set up a coordinate system with the origin at the midpoint of the base of the arch. What are the \(x\) -intercepts of the parabola, and what do they represent? (b) What do the \(x\)- and \(y\)-coordinates of the vertex of the parabola represent? (c) Write an expression for the quadratic function associated with the parabola. (Hint: Use the vertex form of a quadratic function and find the coefficient \(a .)\) (d) What is the \(y\)-coordinate of a point on the arch whose (horizontal) distance from the axis of symmetry of the parabola is 100 feet?
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