Problem 51 - Algebra and Trigonometry

Question

Write an equation in standard form of the parabola that has the same shape as the graph of \(f(x)=2 x^{2},\) but with the given point as the vertex. $$ (-10,-5) $$

Step-by-Step Solution

Answer

The equation of the parabola in its standard form is \(f(x) = 2(x+10)^2 - 5\).

Step 1: Identify the Vertex

The given vertex of the parabola is (-10, -5). So, 'h' is -10 and 'k' is -5.

Step 2: Shape of the Parabola

It is stated in the problem that the parabola has the same shape as the graph of \(f(x) = 2x^2\), which means the new parabola will also be opening either upwards or downwards along the y-axis and will be narrower or wider depending upon the value of 'a'. Here 'a' value is same as that of given parabola i.e., 2.

Step 3: Write the equation

Now substitute the values of 'a', 'h', and 'k' into the general equation of a parabola. The equation is \(f(x) = a(x-h)^2 + k\). Substituting the given and determined values, we get \(f(x) = 2(x-(-10))^2 - 5\).

Step 4: Simplify the equation

The equation could be simplified further to give a more neat representation. Simplifying the equation, we get \(f(x) = 2(x+10)^2 - 5\) which is the equation of a parabola in standard form with the vertex at (-10,-5), and this should be the final solution.

Key Concepts

Standard Form Equation

In mathematics, the standard form equation of a parabola is one of the fundamental ways to represent quadratic functions. Usually, it appears as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. This standard form allows us to easily determine the parabola's direction and its shape. It tells you much about the parabola:

• \(a\) determines whether the parabola opens upwards (if \(a > 0\)) or downwards (if \(a < 0\)).
• The vertex of the parabola can be derived using the formula \(h = \frac{-b}{2a}\) and \(k = f(h)\).

Writing an equation in standard form involves rearranging all elements into this tidy sum, which makes further analysis and transformations relatively straightforward.

Vertex Form

The vertex form of a parabola is quite handy for identifying elements of its graph easily. It's typically expressed as \(f(x) = a(x-h)^2 + k\). As you might guess, \((h, k)\) is the vertex of the parabola. This format is especially useful to:

• Quickly locate the vertex without additional calculations.
• Easily determine the direction of the parabola based on the value of \(a\).

When given a vertex like \((-10, -5)\), the values \(h = -10\) and \(k = -5\) fit directly into the vertex form equation. Substituting these values helps structure the parabola's equation for further processing or conversion into standard form.

Vertex of a Parabola

The vertex is the peak or trough point of a parabola and is central to its graph. Understanding its role helps in visualizing the parabola. When analyzing quadratic functions, remember:

• If \(a > 0\), it's the lowest point (a minimum). If \(a < 0\), it's the highest point (a maximum).
• The coordinates of the vertex, \( (h, k) \), define this point unequivocally.

In practical exercises, you're often given the vertex or asked to find it by transforming the standard form into the vertex form. Recognizing the vertex's pivotal role in the function's graphical representation enriches the understanding of quadratic expressions.

Quadratic Functions

Quadratic functions form the family of second-degree functions and play an essential role in algebra and calculus. Their general expression is \( f(x) = ax^2 + bx + c \). These functions:

• Create parabolic graphs, either opening upwards or downwards depending on \(a\).
• Display symmetry around their vertex, a critical point in their graph.

Through transformations, the standard form, vertex form, and factored form provide diverse ways to analyze these functions' behaviors. Often, you'll translate between these forms to simplify solving specific problems or graphing the equation. By mastering these flexible expressions, navigating through mathematical challenges becomes more manageable.