Problem 49 - 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. $$ (5,3) $$

Step-by-Step Solution

Answer

The equation in standard form of the parabola that has the same shape as the graph of \(f(x)=2 x^{2}\), but with the vertex at (5,3) is \(f(x) = 2x^{2} - 20x + 28\).

Step 1: Write down the vertex form

Start by writing down the vertex form of a parabola, \(f(x) = a(x-h)^{2} + k\). Here, a, h and k are constants. 'a' is the same as in the given parabola equation, 'h' and 'k' are the x and y coordinates of the vertex, respectively.

Step 2: Substitute the values of a, h and k

Now substitute \(a = 2\), \(h = 5\) and \(k = 3\) into the vertex form. This gives the equation \(f(x) = 2(x-5)^{2} + 3\).

Step 3: Convert to standard form

Expand the right side of the equation to convert it to standard form, which is usually more useful for graphing. This simplifies to \(f(x) = 2x^{2} - 20x + 28\).

Key Concepts

Standard Form of a Parabola

The standard form of a parabola is expressed as \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. This form is particularly useful in identifying the parabola's vertical direction and making it easier to use in graphing. The coefficient \(a\) determines if the parabola opens upwards \((a > 0)\) or downwards \((a < 0)\).
Understanding the structure:

• \(ax^2\) tells us the width and direction of the parabola.
• \(bx\) affects the symmetry and shift along the x-axis.
• \(c\) is the y-intercept, indicating where the parabola crosses the y-axis.

For example, converting the vertex form \(f(x) = 2(x-5)^2 + 3\) to standard form involves expanding it into \(f(x) = 2x^2 - 20x + 28\). This process includes distributing the \(a\) term across the squared binomial and simplifying.

Vertex Form of a Parabola

The vertex form's equation is \(f(x) = a(x-h)^2 + k\), where \((h, k)\) represents the vertex of the parabola. This form is incredibly valuable when you want to quickly identify the vertex of the parabola. Knowing the parabola’s peak or lowest point ( for upward-opening parabolas) is immensely helpful in graphing situations.
Key components to watch for:

• \((h, k)\) denotes the vertex's x and y coordinates.
• \('a'\) dictates the direction it opens and its width, just like in the standard form.

When a parabola's vertex is given, as in our example where the vertex is \((5, 3)\), substituting \(a = 2\), \(h = 5\), and \(k = 3\) into the vertex form yields \(f(x) = 2(x-5)^2 + 3\). This form directly shows where the vertex is located without any algebraic manipulation.

Graphing Parabolas

Graphing parabolas can be an exciting challenge as it brings algebraic expressions to life as curves on the coordinate plane. A sound understanding of both the standard and vertex forms aids in this process.
Here's how you can approach graphing a parabola:

• Identify the vertex and plot it. For example, in the vertex form \(f(x) = 2(x-5)^2 + 3\), the vertex is \((5, 3)\).
• Determine the direction of the parabola using the \(a\) value. If \(a > 0\), it opens upwards; if \(a < 0\), downwards.
• Use the standard form to find the y-intercept by setting \(x = 0\) and solve for \(f(x)\). In the example \(f(x) = 2x^2 - 20x + 28\), the y-intercept is at 28.
• Draw the axis of symmetry, a vertical line through the vertex, aiding in reflecting points across this line for accurate graphing.

By combining these steps together, you can better visualize, sketch, and understand how the algebraic expressions translate into real-world graphs.