Problem 48 - Algebra A Combined Function

Question

Write the equation of the parabola that has the same shape as \(f(x)=5 x^{2}\) but with the given vertex. Call each function \(g(x) .\) $$ (1,6) $$

Step-by-Step Solution

Answer

The equation of the parabola is \( g(x) = 5x^2 - 10x + 11 \).

Step 1: Understand the Standard Form

The standard form of a parabola equation is \( g(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex and \(a\) determines the shape. We are given that the parabola should have the same shape as \( f(x) = 5x^2 \), which means \(a = 5\).

Step 2: Substitute the Vertex into the Formula

Substitute the vertex \((1, 6)\) into the standard form equation. This gives us \( g(x) = 5(x - 1)^2 + 6 \). Here, \(h = 1\) and \(k = 6\).

Step 3: Simplify the Expression

Expand the equation \( g(x) = 5(x - 1)^2 + 6 \).First, calculate \((x - 1)^2\): \( (x - 1)^2 = x^2 - 2x + 1 \).Then, substitute back into the equation:\( g(x) = 5(x^2 - 2x + 1) + 6 \).

Step 4: Distribute \(5\) Across the Terms Inside Parentheses

Multiply each term inside the parentheses by \(5\):\[ g(x) = 5x^2 - 10x + 5 + 6 \].

Step 5: Combine Like Terms

Combine the constant terms in the expanded equation:\[ g(x) = 5x^2 - 10x + 11 \].

Key Concepts

Vertex Form of a Parabola

The vertex form of a parabola is a very useful format that makes it easy to identify the vertex of the parabola. It is expressed as \[ g(x) = a(x - h)^2 + k \]where

• \(a\) determines the width and direction of the parabola (whether it opens upwards or downwards),
• \((h, k)\) is the vertex.

By knowing the vertex, you can quickly sketch the graph of the parabola. The vertex is a crucial point as it is either the maximum or minimum point on the graph, depending on whether the parabola opens downwards or upwards respectively. In the given problem, the vertex is at \((1, 6)\), making it easy to substitute into the vertex form.

Expanding Quadratic Expressions

Expanding quadratic expressions involves expressing a squared binomial as a trinomial. Let's consider the expression \((x-1)^2\). This can be expanded to \(x^2 - 2x + 1\).Here's how it works:

• Take the first term and square it: \(x^2\).
• Multiply the first and second terms, and double the result: \(-2x\).
• Square the second term: \(1\).

Combine these to get the trinomial. Expanding expressions is a fundamental algebraic skill, especially useful in transforming vertex form equations into standard polynomial form. This process helps in various mathematical operations such as factoring, solving equations, and graphing.

Transformations of Functions

Transformations allow us to shift, stretch, and manipulate functions in various ways. Understanding transformations of parabolas is crucial as it enables us to modify the graph into a desired position or shape. The equation \(g(x) = a(x - h)^2 + k\) involves several transformations:

• The \(h\) value shifts the parabola horizontally. If \(h\) is positive, shift to the right; if negative, shift to the left.
• The \(k\) value moves the parabola vertically. Positive \(k\) value shifts it up, while negative \(k\) takes it down.
• The \(a\) value impacts the opening's width and direction. A larger \(|a|\) makes the parabola narrower, while a smaller \(|a|\) widens it.

By grasping these transformations, you can effectively graph and understand how functions behave under different conditions.

Combining Like Terms

Combining like terms is an essential step in simplifying expressions. This occurs when you have terms in an expression that share the same variable raised to the same power. For example, in the expanded equation \(g(x) = 5x^2 - 10x + 5 + 6\), the constants \(5\) and \(6\) are like terms.To combine like terms:

• Identify the terms with identical variables and exponents.
• Add or subtract the coefficients of these terms as appropriate.

Thus, combining \(5\) and \(6\) yields \(11\), giving us the final simplified form: \(g(x) = 5x^2 - 10x + 11\). Simplifying expressions in this way helps clarify and streamline complicated expressions, making them easier to interpret and solve.