Problem 47 - Algebra 2

Question

For each function, the vertex of the function's graph is given. Find \(c .\) $$ y=c-x^{2}-2 x ;(-1,3) $$

Step-by-Step Solution

Answer

The value of \(c\) for the given function with vertex \((-1, 3)\) is \(2\).

Step 1: Identify the Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by the formula \(y=a(x-h)^2+k\) where \((h,k)\) is the vertex of the parabola. Here, we need to rewrite the given function \(y =c-x^{2}-2x\) in order to clearly identify its vertex.

Step 2: Express the Given Function in Vertex Form

To express the given function in vertex form, we need to complete the square. We can do this by reorganizing and factoring the quadratic expression as follows: \(y = c - (x^2 + 2x)\). Let's complete the square by adding and subtracting the square of half the coefficient of \(x\), which is \((2/2)^2=1\). Thus, we have \(y = c - [x^2 + 2x + 1 - 1] = c - [(x + 1)^2 - 1]\). This gives us \(y = (c + 1) - (x + 1)^2\).

Step 3: Match the Vertex to the Function

The vertex of the function is given as \((-1, 3)\). Using the form \(y = (c + 1) - (x + 1)^2\), we can substitute \(x=-1\) and \(y=3\) to find the value of \(c\). This yields \(3 = (c + 1) - (0)^2\), which simplifies to \(3 = c + 1\).

Step 4: Solve for the Value of \(c\)

We can now solve for \(c\) by subtracting 1 from both sides of the equation \(3 = c + 1\). Subtracting gives us \(c = 3 - 1 = 2\). Therefore, we have found that \(c = 2\).

Key Concepts

Completing the Square

In the context of quadratic functions, completing the square is a method used to rewrite a quadratic expression into a perfect square trinomial, which simplifies the process of finding the function's vertex or solving the equation. The process involves adding and subtracting a particular value inside the quadratic expression. To illustrate this, let's look at a quadratic expression like y = ax^2 + bx + c. The coefficient 'a' must be 1 before completing the square, if not, divide the entire equation by 'a' first.

The steps involved in completing the square are:

• Rearrange the equation to place the constant term 'c' on the other side of the equals sign.
• Take half of the coefficient of 'x' (which is 'b') and square it.
• Add and subtract this squared number within the parentheses of the quadratic term.
• Factor the perfect square trinomial created, and rewrite the equation as y = a(x - h)^2 + k.

Completing the square is essential as it helps in solving quadratic equations and allows us to clearly see the vertex form of a quadratic function.

Quadratic Equations

Quadratic equations are fundamental in algebra and represent curves called parabolas when graphed. They have the standard form ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants. Unlike linear equations with a single solution, a quadratic equation usually has two solutions which can be real or complex numbers.

These equations can be solved by various methods including:

• Factoring, if the quadratic is factorable.
• Using the quadratic formula: x = [-b ± sqrt(b^2 - 4ac)] / (2a).
• Graphically, by find the points where the parabola crosses the x-axis.
• Completing the square, which transforms the equation into a perfect square form making it easier to solve.

Familiarity with quadratic equations is not just important for solving algebra problems, but also for understanding phenomena in physics and other sciences that are modeled by quadratic relationships.

Vertex of a Parabola

The vertex of a parabola is a crucial concept as it represents the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards respectively. For the quadratic function in vertex form y = a(x - h)^2 + k, the vertex is the point (h, k).

When looking at the graph of a quadratic function, the vertex is the turning point, which means it is the point where the function changes direction.

• If 'a' is positive, the parabola opens upwards, and the vertex is the minimum point.
• If 'a' is negative, the parabola opens downwards, and the vertex is the maximum point.

Determining the vertex is not only important for graphing the parabola but also for solving optimization problems where you need to find the maximum or minimum values of a quadratic function.