Problem 62 - 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 (2,3) and has the shape of \(f(x)=3 x^{2}\). Vertex is on the \(y\) - axis.

Step-by-Step Solution

Answer

The equation is \(g(x) = 3x^2 - 9\).

Step 1: Identify the Generic Form

Given that the function has a shape similar to \(f(x) = 3x^2\), we know the quadratic function is in the form \(g(x) = 3(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola.

Step 2: Determine Vertex Position

Since the vertex is on the y-axis, this means \(h = 0\). Therefore, the equation simplifies to \(g(x) = 3x^2 + k\).

Step 3: Use Given Point to Find k

Substitute the point \((2, 3)\) into the equation: \[3 = 3(2)^2 + k\]. Solve for \(k\):\[3 = 12 + k\]This simplifies to \(k = -9\).

Step 4: Write the Final Equation

Now that we have \(k = -9\), substitute back into the equation: \(g(x) = 3x^2 - 9\). This is the quadratic function that contains the point \((2, 3)\) and has the same shape as \(f(x) = 3x^2\) with the vertex on the y-axis.

Key Concepts

Vertex Form of a Quadratic Function

Understanding the vertex form of a quadratic function is crucial when working with parabolas. This form allows us to easily determine the shape and position of the parabola, as well as solve related equations efficiently. The vertex form of a quadratic function is expressed as:

• \(f(x) = a(x - h)^2 + k\)

where:

• \(a\) is a non-zero constant that stretches or compresses the graph vertically.
• \((h, k)\) is the vertex of the parabola, a point that represents the 'turning' point of the parabola.

By this format, the value \(h\) determines the horizontal translation, while \(k\) adjusts the vertical translation of the parabola along the axes. This form is especially helpful when compared to the standard form \(ax^2 + bx + c\) since it directly communicates the position of the vertex without needing to complete the square.

Parabola Vertex

The vertex of a parabola is a key point that defines its position and direction. It can be seen as the 'peak' or 'trough' of the U-shaped graph. By utilizing the vertex form of a quadratic function, you can easily identify this important point.For parabolas that open upward or downward, the vertex is where the direction of the curve switches, marking the highest or lowest point respectively.

• An upward-facing parabola has its vertex as the minimum point.
• A downward-facing parabola has its vertex as the maximum point.

This information becomes handy in optimization problems or when determining the range of the function.Moreover, in cases where the vertex is on the y-axis, like in our exercise with \(h = 0\), the process of finding the function simplifies as it removes the horizontal shift component.

Solving Quadratic Equations

Solving quadratic equations is a fundamental skill in algebra, often involving different techniques depending on the form of the quadratic function. One common method is using the quadratic formula, while for certain situations, the vertex form makes solving more intuitive.To solve a quadratic equation, the task generally involves finding the x-values that make the function equal to zero (roots of the equation). For equations in vertex form, you can:

• Rearrange the equation to isolate the squared term.
• Take the square root of both sides, considering both the positive and negative roots.

This approach is often simpler because it leverages the structure of the vertex form. The availability of the vertex makes calculations around translations and stretching much more straightforward.In our exercise, the equation was tailored by using a known point on the parabola to find the unknown \(k\), hence creating a precise match to the given features.