Problem 65 - College Algebra with Corequisite Support

Question

For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function. Contains (1,-6) has the shape of \(f(x)=3 x^{2}\). Vertex has \(\mathrm{x}\) coordinate of -1.

Step-by-Step Solution

Answer

The quadratic function is \( y = 3(x + 1)^2 - 18 \).

Step 1: Identify the forms

The given function is \( f(x) = 3x^2 \), which is in the standard form of a quadratic equation \( ax^2 + bx + c \). Since the new function needs to have the same shape, its equation will be of the form \( y = 3(x - h)^2 + k \).

Step 2: Use vertex form

The vertex form of the quadratic equation is \( y = a(x - h)^2 + k \). We know the vertex has \( x \) coordinate \( h = -1 \) and \( a = 3 \). So, the quadratic equation becomes \( y = 3(x + 1)^2 + k \).

Step 3: Solve for k using the point

The quadratic function needs to pass through the given point \((1, -6)\). Substitute \( x = 1 \) and \( y = -6 \) into the equation \( y = 3(x + 1)^2 + k \). This gives us \( -6 = 3(1 + 1)^2 + k \) or \( -6 = 3(2)^2 + k \).

Step 4: Simplify and solve for k

Solve \( -6 = 3(4) + k \). This simplifies to \( -6 = 12 + k \). Subtract 12 from both sides to find \( k = -18 \).

Step 5: Write the final equation

Using the values for \( a \), \( h \), and \( k \), the quadratic function is \( y = 3(x + 1)^2 - 18 \).

Key Concepts

Vertex Form

The vertex form of a quadratic function is a way of expressing the equation that highlights the vertex, or the highest or lowest point on the graph of the function.
In vertex form, a quadratic function is written as \( y = a(x - h)^2 + k \), where:

• \( a \) indicates the opening direction and width of the parabola
• \( h \) is the x-coordinate of the vertex
• \( k \) is the y-coordinate of the vertex

This format is particularly useful when you need to identify the vertex easily. For example, with \( y = 3(x + 1)^2 - 18 \), the vertex would be at \( (-1, -18) \).
Vertex form is excellent for graphing, as it moves the standard parabola \( y = x^2 \) over to \( (h, k) \) and stretches it according to \( a \).

Solving Quadratic Equations

Solving quadratic equations is a fundamental aspect of algebra that involves finding the values of \( x \) that make the equation true.
There are several methods to solve these equations, including:

• Factoring: Used when the quadratic can be expressed as a product of two linear expressions.
• Completing the Square: This method involves reorganizing the quadratic into vertex form.
• Quadratic Formula: A universal method used for any quadratic equation, given as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Graphing: Offers a visual approach to finding the roots of the equation, where the graph intersects the x-axis.

In our example, solving for \( k \) using the point (1, -6) involves substituting these coordinates into the vertex form and calculating to find \( k = -18 \). This demonstrates using specific points to solve the equation.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is represented as \( ax^2 + bx + c \), where:

• \( a \), \( b \), and \( c \) are constants
• \( a \) indicates the direction and width of the parabola
• \( b \) affects the symmetry and position of the parabola on the graph
• \( c \) is the y-intercept, where the parabola crosses the y-axis

This form is prevalent in basic algebraic manipulations such as expansion and simplification.
For example, the given quadratic function \( f(x) = 3x^2 \) is already in standard form with \( a = 3 \), and both \( b \) and \( c \) equal to 0. This form does not reveal the vertex directly but provides a good starting point for converting to other forms, like the vertex form, for different tasks such as graphing or solving. From this format, you can also easily derive properties like the axis of symmetry and whether the parabola opens upwards or downwards.