Problem 42 - Algebra 2

Question

A parabola contains the points \((0,-4),(2,4),\) and \((4,4) .\) Find the vertex.

Step-by-Step Solution

Answer

The vertex is at \((1, -5)\).

Step 1: Determine the quadratic function

Use the point-slope form to construct a system of equations based on the general quadratic equation for a parabola opening upward or downward, which is of the form y = ax^2 + bx + c, using the given points (0, -4), (2, 4), and (4, 4).

Step 2: Solve the system of equations

Substitute the coordinate pairs into the general equation to obtain three equations: 1) Substituting (0,-4) gives us c = -4. 2) Substituting (2,4) gives us 4a + 2b - 4 = 0. 3) Substituting (4,4) gives us 16a + 4b - 4 = 0. These equations can then be solved simultaneously to find the values of a and b.

Step 3: Calculate values of a and b

After finding c = -4 from Equation 1, we use Equations 2 and 3 to calculate a and b. Using substitution or the elimination method can be helpful to solve for these coefficients.

Step 4: Identify the parabola's vertex

The vertex form of a parabola is y = a(x - h)^2 + k, where (h, k) is the vertex. Using the coefficients a and b, we can determine the vertex by the formula h = -b/(2a). Substitute a and b to find the x-coordinate of the vertex. The y-coordinate is found by then plugging \(h\) back into the equation.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial equation that is of the second degree, typically written in the form of \( y = ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). The graph of a quadratic function is known as a parabola, which can either open upwards or downwards. This is determined by the sign of the coefficient \(a\): if \(a > 0\), the parabola opens upward, and if \(a < 0\), it opens downward.

The shape and location of the parabola are influenced by \(a\), \(b\), and \(c\). The vertex of the parabola represents the highest or lowest point on the graph, depending on the direction the parabola opens. To find the parabola that contains specific points, such as \( (0, -4), (2, 4), (4, 4) \), we plug these points into the general quadratic equation to create a system of equations.

Point-Slope Form

The point-slope form of a line is an equation that enables one to write a line equation when one knows the slope and a point on the line. This is not to be confused with the quadratic equation, although it can sometimes be used as a stepping stone when working with quadratic functions. The point-slope form is given by \( y - y_1 = m(x - x_1) \), where \(m\) is the slope and \( (x_1, y_1) \) are the coordinates of the given point. In the context of solving quadratic equations, this form is less commonly applied directly. However, understanding the concept of how equations can be manipulated based on known points and slopes is valuable when working with parabolas.

System of Equations

A system of equations is a set of two or more equations that share common variables. In the case of a quadratic function, when we have multiple points that lie on a parabola, we can substitute these points into the general quadratic equation to get a system of equations. Each point yields one equation and substitution of the coordinate pair \( (x, y) \) results in a single equation.

With the example problem, we obtain the following system with three unknowns, \(a\), \(b\), and \(c\), and three equations to solve for them. Solving this system gives the specific values for the coefficients of the quadratic function that will model the parabola passing through the given points. It is also essential to use consistent methods, like the elimination method or substitution, to solve these equations accurately.

Vertex Form of a Parabola

In the context of a parabola, the vertex form is invaluable as it provides clear information about the graph's peak or trough. The vertex form of a parabola is written as \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola. This form makes it very simple to identify the vertex, as the coordinates of the vertex are directly given by \(h\) and \(k\).

Once we have solved for the coefficients \(a\), \(b\), and \(c\) using the system of equations derived from given points on the parabola, we can convert the general form into the vertex form. This conversion can involve completing the square or using formulas derived from the coefficients of the quadratic function. The vertex form is particularly useful for graphing and analyzing the properties of the parabola.

Elimination Method

The elimination method is a systematic way of solving a system of equations where you eliminate one or more variables by adding or subtracting equations from each other. This is especially useful when dealing with a system of equations from a quadratic function.

To apply the elimination method, one must align terms with common variables and manipulate the equations to cancel out one of the variables. For instance, if we have the equations \(4a + 2b - 4 = 0\) and \(16a + 4b - 4 = 0\), we can multiply the first equation by (-2) and add it to the second equation to eliminate \(b\) and solve for \(a\). Once \(a\) is found, we substitute it back into either equation to find \(b\). This method is particularly helpful when dealing with multiple unknowns as it helps isolate and solve for the unknowns systematically.