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Chapter 11: Problem 90

For each of the following, write the equation of the parabola that has the shape of \(f(x)=2 x^{2}\) or \(g(x)=-2 x^{2}\) and has a maximum value or \(a\) minimum value at the specified point. Minimum: \((-4,0)\)

Short Answer

Expert verified
\(y = 2(x + 4)^2\)

Step by step solution

01

Determine the Vertex Form of the Parabola

The vertex form of a parabola's equation can be written as \[ y = a(x - h)^2 + k \] where \( (h, k) \) is the vertex of the parabola. Here, \(a\) indicates whether the parabola opens upwards (a > 0) or downwards (a < 0).
02

Identify the Vertex

Given that the parabola has a minimum value at \((-4, 0)\), the vertex \( (h, k) \) of the equation is \( (-4, 0) \).
03

Determine the Equation Based on the Shape

Since the parabola has a shape similar to \( f(x) = 2x^2 \) and opens upwards (a > 0), we use \(a = 2\). Thus, the parabola's equation should be: \[ y = 2(x + 4)^2 \].
04

Write the Final Equation

Given all the information, the equation of the parabola is \[ y = 2(x + 4)^2 \].

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Key Concepts

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

Vertex Form
The vertex form of a quadratic equation is a special way to write parabolas and has the format:




\[y = a(x - h)^2 + k \]
Here,
  • This form makes it clear where the vertex of the parabola is located.
    In the equation,
  • k represents the y-coordinate of the vertex
  • giving us the point
  • \[(h, k)\].
    This form is very useful because the vertex is crucial for understanding the behavior of the quadratic function. By adjusting 鈥渁 鈥�(positive or negative), .
Minimum Value
The minimum value of a quadratic function is the smallest y-value the function can attain. For a parabola that opens upwards, this point is at the vertex.
In vertex form, the minimum value of the parabola occurs when

\( y = k \)
- That's because
The term \(a(x - h)^2 \) (non-negative).
In our example, because the vertex is at \br>the y-coordinate (k) is 0, so the minimum value of the function is 0.
Quadratic Function
A quadratic function is one that can be written in the form: \[y = ax^2 + bx + c\]
Quadratic functions create parabolas when graphed. They have a squared term which gives the characteristic 'U' shape known as a parabola. Some key features of quadratic functions are:
  • They are symmetrical about a vertical line known as the axis of symmetry
    • ...

... The vertex form is especially handy for reading off the vertex and understanding the parabola's shape more easily.
Parabola Opens Upwards
Whether a parabola opens upwards or downwards depends on the coefficient 'a' in the quadratic function.
  • If 'a' is positive, the parabola opens upwards.

  • If 'a' is negative, the parabola opens downwards.


    • Opening upwards means the parabola has a minimum point at its vertex, and the arms of the parabola extend upwards to positive infinity.
    In our solution, with \(a = 2\),
    , we know the parabola opens upwards...

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    Most popular questions from this chapter

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    Complete the square to find the \(x\) -intercepts of each function given by the equation listed. $$ f(x)=x^{2}-8 x-10 $$

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