Problem 61 - 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 (-1,4) and has the shape of \(f(x)=2 x^{2}\). Vertex is on the \(y\) - axis.

Step-by-Step Solution

Answer

The quadratic function is \(g(x) = 2x^2 + 2\).

Step 1: Understanding the Problem

We need to find a quadratic function that passes through the point \((-1, 4)\) and retains the same shape as the given function \(f(x) = 2x^2\). Since the vertex is on the \(y\)-axis, the function is in the form \(g(x) = 2x^2 + c\).

Step 2: Substitute Point into Function

Since the function must pass through \((-1, 4)\), substitute \(x = -1\) and \(y = 4\) into the equation \(g(x) = 2x^2 + c\). This gives: \[4 = 2(-1)^2 + c.\]

Step 3: Solve for Constant c

Simplify the equation \[4 = 2(1) + c\] to \[4 = 2 + c.\] Solve for \(c\) by subtracting 2 from both sides: \[c = 2.\]

Step 4: Write the Function Formula

Substitute \(c = 2\) back into the function \(g(x) = 2x^2 + c\). The quadratic function becomes \(g(x) = 2x^2 + 2\).

Key Concepts

Vertex Form

When dealing with quadratic functions, the vertex formis a specific way to express these functions. It makes it super easy to identify important features of the graph, such as the vertex, which is the "peak" or "lowest" point (in case of a parabola opening upwards).

For a quadratic function, the vertex form is generally given by:\[ g(x) = a(x - h)^2 + k \]Here:

• The vertex is represented by the point \((h, k)\).
• The letter \(a\) denotes the same shape and direction as the parabola in standard form.
• Setting \(h = 0\) indicates the vertex lies on the \(y\)-axis.

Our exercise involves identifying the quadratic function, which is expressed as \[ g(x) = 2x^2 + c \]indicating its symmetry to the \(y\)-axis, making it an example of a vertically shifted quadratic. As depicted by this form, if you know the value of \(c\), you can effortlessly determine the full expression of the function based on a given point.

Point Substitution

Point substitution is a handy method to find specific constants in the equation of a function when a point is given. The idea is simple: place the known coordinates into the function to solve for unknowns or adjust the function to suit given conditions.

In our problem, we know the function must pass through a specific point, \((-1, 4)\). Given this information, we substitute these values into the quadratic equation:\[ 4 = 2(-1)^2 + c \]This substitution means that wherever you see \(x\), you use \(-1\), and where there is \(g(x)\), use 4.

• \(4\) is our \(y\)-value from the point, matching the output of the function.
• \(-1\) is the \(x\)-value from the point.
• The unknown \(c\) is what we solve for in this case.

By substituting these values, we turn what initially seemed like a complex equation into a simple arithmetic problem. By solving, we found \(c = 2\). This process is essential for modifying functions so they conform to specified conditions.

Quadratic Equations

Quadratic equations are polynomials of degree two. They are typically written in the standard form:\[ ax^2 + bx + c = 0 \]Here, the variable is usually \(x\), and \(a\), \(b\), and \(c\) are constants, with \(a eq 0\).

The quintessential property of quadratic functions is their shape - they form a parabola when graphed. This parabola can open upwards or downwards, depending on the sign of \(a\). The vertex is the tip of this U-shaped curve.

In our exercise, we are tasked to form a specific quadratic function, leveraging a known shape from the given function \(f(x) = 2x^2\) ensuring the same "stretch" and direction.

• The coefficient \(2\) implies both have the same rise rate (up or down).
• Our modified function becomes \(g(x) = 2x^2 + 2\) after adjustments, still a quadratic function.
• This updated quadratic precisely represents our conditions: passing through the point and maintaining the original shape.

Understanding quadratic equations is fundamental as they frequently appear in different math problems, serving as fundamental building blocks for more complex explorations.