Problem 30 - Algebra for College Students

Question

Find the vertex for the graph of each quadratic function. $$f(x)=x^{2}+12$$

Step-by-Step Solution

Answer

The vertex is (0, 12).

Step 1: Identify the standard form

A quadratic function can be written in the standard form as \[f(x) = ax^2 + bx + c\]. For the given function \[f(x) = x^2 + 12\], identify the coefficients: \[a = 1\], \[b = 0\], and \[c = 12\].

Step 2: Use the vertex formula

The vertex of a quadratic function in standard form can be found using the vertex formula \[x = -\frac{b}{2a}\]. Substitute \[a = 1\] and \[b = 0\] into the formula: \[x = -\frac{0}{2 \cdot 1} = 0\].

Step 3: Find the y-coordinate of the vertex

Once the x-coordinate of the vertex is found, substitute it back into the original equation to find the y-coordinate. \[f(0) = (0)^2 + 12 = 12\].

Step 4: Write the vertex

Therefore, the vertex of the function \[f(x) = x^2 + 12\] is \[(0, 12)\].

Key Concepts

quadratic function

A quadratic function is a type of polynomial that has the form \[f(x) = ax^2 + bx + c\]. In this formula, \[a, b,\] and \[c\] are constants, and \[x\] is the variable. The graph of a quadratic function is a curve called a parabola, which can open upwards or downwards depending on the sign of \[a\].
Some key characteristics of quadratic functions include:

• The highest power of \[x\] is 2.
• The graph is a parabola.
• The vertex is the highest or lowest point on the graph, depending on whether the parabola opens downwards or upwards.

Understanding the shape and properties of quadratic functions is crucial for solving a wide range of mathematical problems, including those in physics, engineering, and economics. Now, let's dive into the topics of the vertex formula and the standard form of quadratic functions.

vertex formula

The vertex of a quadratic function is a point where the parabola either reaches its minimum or maximum value. Finding the vertex is important because it tells us the vertex's location on the graph. To find the x-coordinate of the vertex for a quadratic function in standard form \[f(x) = ax^2 + bx + c\], we use the vertex formula \[x = -\frac{b}{2a}\].
This formula works by taking the coefficients \[a\] and \[b\] from the quadratic equation and plugging them directly into the formula. For example, in the function \[f(x) = x^2 + 12\], we have \[a = 1\] and \[b = 0\]. Using the vertex formula, we get:
\[x = -\frac{0}{2 \cdot 1} = 0\]
Once the x-coordinate is found, we substitute it back into the original function to find the y-coordinate. This makes the vertex complete, pinpointing the exact highest or lowest point on the curve. In our case,
\[f(0) = (0)^2 + 12 = 12\]
Thus, the vertex is at \[ (0, 12) \].

standard form of quadratic

The standard form of a quadratic function is written as \[f(x) = ax^2 + bx + c\]. This form is very useful for various mathematical processes, including finding the vertex and graphing the parabola.
Here are some key points about the standard form:

• The coefficient \[a\] determines the direction of the parabola. If \[a > 0\], it opens upwards. If \[a < 0\], it opens downwards.

• The coefficient \[b\] affects the position of the vertex along the x-axis.

• The coefficient \[c\] represents the y-intercept, where the graph crosses the y-axis.

Let's apply this to our example \[f(x) = x^2 + 12\], which is already in standard form:

• Here, \[a = 1\], which means the parabola opens upwards.

• Since \[b = 0\], the vertex's x-coordinate is right on the y-axis.

• With \[c = 12\], the parabola crosses the y-axis at the point (0, 12).

Because this function is simple, finding the vertex is straightforward and helps in understanding how the parabola behaves.