Problem 79 - Elementary Algebra

Question

Rewrite in \(y=a(x-h)_{2}+k\) form and determine the vertex. $$ y=x_{2}-14 x+24 $$

Step-by-Step Solution

Answer

The equation in vertex form is \(y=(x-7)^2-25\) with vertex (7, -25).

Step 1: Identify the Expression

We start by identifying the initial quadratic expression: \[ y = x^2 - 14x + 24 \] Our goal is to rewrite this in vertex form, which is given by: \[ y = a(x-h)^2 + k \] where \(h, k\) is the vertex.

Step 2: Complete the Square

To rewrite this expression in vertex form, we complete the square. Start by looking at \[ x^2 - 14x \] Add and subtract the square of half the coefficient of x:Half of \(-14\) is \(-7\), so we add and subtract \(7^2 = 49\):\[ x^2 - 14x + 49 - 49 + 24 \] becomes \[ (x-7)^2 - 49 + 24 \] which simplifies to \[ (x-7)^2 - 25 \].

Step 3: Write in Vertex Form

Now combine the previous expression to fit the vertex form:\[ y = (x-7)^2 - 25 \] This matches the vertex form \(y = a(x-h)^2 + k\), with \(a = 1\), \(h = 7\), and \(k = -25\).

Step 4: Identify the Vertex

From the vertex form \(y = (x-7)^2 - 25\), we see that the vertex \((h, k)\) can be directly read off as:Vertex: \((7, -25)\).

Key Concepts

Vertex Form

The vertex form of a quadratic equation is a way to express the equation making it easy to identify the vertex of the parabola it represents. Quadratic equations are usually given in standard form like \[ y = ax^2 + bx + c. \] The vertex form rearranges this equation to focus on the vertex of the parabola, which is \[ y = a(x-h)^2 + k, \] where \((h, k)\) is the vertex of the parabola.
Why use vertex form? There are a few excellent reasons:

• It makes it very easy to identify the vertex.
• It shows how the parabola is shifted from the origin.
• It makes graphing the equation more intuitive.

To convert from standard form to vertex form, we often use a process called completing the square. This gives us insight into how shifts in \(x\) and \(y\) positions affect the graph, which makes solving equations and graphing much simpler.

Completing the Square

Completing the square is a technique used to convert a quadratic equation in standard form into vertex form. It's a step-by-step process that reshapes the expression so that it can be easily rewritten into a perfect square plus or minus some number. Let's break this down:We start with the quadratic term and the linear term \(x^2 - 14x\) and our goal is to add a constant to make it a perfect square trinomial. Here's how:

• Take half of the linear coefficient (the \(-14\) here), which is \(-7\), and square it. This gives \(49\).
• Add \(49\) to \(x^2 - 14x\) to form \((x - 7)^2\), if we add inside we also have to subtract it to keep the equation balanced.
• The original equation \(y = x^2 - 14x + 24\) transforms into \(y = (x-7)^2 - 49 + 24\).

By completing the square, what would have been a difficult-to-visualize expression becomes a clear format, \((x - 7)^2\), descriptive of a shift and stretch of the parabola.

Vertex Identification

After converting a quadratic equation into vertex form, identifying the vertex becomes straightforward. In the expression \(y = a(x-h)^2 + k,\) we recognize that \((h, k)\) is the vertex of the parabola. So, once we've successfully rewritten our quadratic equation, identifying these values gives us precise insights into its graph's behavior.
For instance, after completing the square, the equation \(y = (x-7)^2 - 25\) clearly shows that the vertex is at \((7, -25)\). This means:

• The parabola shifts 7 units to the right along the x-axis from the origin.
• It shifts 25 units down along the y-axis.

With these values, not only can we pinpoint exactly where the parabola reaches its maximum or minimum value, but we can also graph the equation easily. This makes vertex identification an essential tool in analyzing and utilizing quadratic functions.