Problem 44 - Algebra 2

Question

For each function, the vertex of the function's graph is given. Find \(c .\) $$ y=x^{2}-6 x+c ;(3,-4) $$

Step-by-Step Solution

Answer

The value of \(c\) is 5.

Step 1 - Identifying the Vertex Formula

The vertex of a parabola given by the equation in vertex form, which is \(y = a(x-h)^2 + k\), is the point \((h, k)\). For the standard form equation \(y = ax^2 + bx + c\), the x-coordinate of the vertex is given by \(-\frac{b}{2a}\). In this case, our given vertex is \((3, -4)\).

Step 2 - Finding the Value of 'c'

To find the value of \(c\), we use the coordinates of the vertex \((3, -4)\) and the equation \(y = x^2 - 6x + c\). Substituting \(x = 3\) and \(y = -4\), we get \(-4 = (3)^2 - 6(3) + c\).

Step 3 - Solving for 'c'

Solve the above equation for \(c\): \(-4 = 9 - 18 + c\). Simplify the right-hand side to get \(c = -4 - 9 + 18\). Calculate the value of \(c\) to obtain \(c = 5\).

Key Concepts

Quadratic Functions

Quadratic functions are foundational in understanding algebra and appear in various applications across disciplines such as physics, engineering, and economics. The general form of a quadratic function is expressed as
\begin{center}\(y = ax^2 + bx + c\)d{center}
where \(a\), \(b\), and \(c\) are constants, with \(a\) not equal to zero.
Quadratic functions are famous for creating a curved graph known as a parabola. Parabolas can either open upwards or downwards, depending on the sign of the coefficient \(a\). If \(a\) is positive, the parabola opens upwards, and if negative, it opens downwards. The highest or lowest point on this graph is called the vertex, and it represents a key feature as it marks the maximum or minimum value of the function.
To optimize solving quadratic equations, sometimes we need to rewrite them into different forms to make certain properties, like the vertex, more apparent. This necessity leads us to explore the standard form and vertex form of quadratic equations.

Standard Form Equation

The standard form equation of a quadratic is perhaps the most well-known representation and is written as
\begin{center}\(y = ax^2 + bx + c\)d{center}
where again, \(a\), \(b\), and \(c\) are constants, and \(x\) represents the independent variable. When the quadratic function is in this format, the coefficient \(a\) indicates the direction and width of the parabola, and \(c\) will give you the y-intercept, or where the graph crosses the y-axis.
The standard form makes it straightforward to use methods like factoring, completing the square, and the quadratic formula to find the x-intercepts or roots of the function. Nevertheless, it does not readily reveal the vertex of the parabola. To find the vertex, we use the formula for the x-coordinate of the vertex, \(h = -\frac{b}{2a}\), and then substitute \(h\) back into the standard form equation to find the corresponding y-coordinate \(k\).
In our textbook exercise, we see the application of these concepts where the given vertex information is utilized to determine the unknown constant \(c\) by substituting the vertex coordinates back into the standard form equation.

Vertex Form of a Quadratic Equation

The vertex form of a quadratic equation is a powerful tool for graphing and analyzing the properties of parabolas. It is written as
\begin{center}\(y = a(x-h)^2 + k\)d{center}
where \(a\) is the same as in the standard form, affecting the width and direction of the parabola. The terms \(h\) and \(k\) represent the coordinates of the vertex of the parabola. Thus, the vertex is easily identified as
\((h, k)\).
The vertex form makes it straightforward to determine translations of the parabola; if \(h\) is positive, the parabola is shifted to the right, if negative, to the left. Likewise, a positive \(k\) shifts the parabola up, and a negative \(k\) shifts it down.
To convert a quadratic equation from standard form to vertex form, one can complete the square, which involves a few steps but ultimately provides a form that gives quick insights into the graph of the function. In the solution provided in the textbook, understanding the vertex form is essential as it directs the student on how to plug in the vertex values to solve for the unknown constant \(c\).