How to Identify Characteristics of Quadratic Functions: Equations

Identifying the characteristics of the quadratic function helps to better understand the behavior of these functions. For this reason, in this article, we are going to study and examine the characteristics of the quadratic function.

Related Topics

• How to Find the Axis of Symmetry of Quadratic Functions?
• Quadratic Function

A step-by-step identify Characteristics of Quadratic Functions: Equations

The standard form of the quadratic function is in the form \(y=ax^2+bx+c\), where \(a\), \(b\), and \(c\) are real numbers and \(a≠0\). Also, the vertex form of the quadratic function is shown as follows:

\(y=a(x – h)^2+k\)

where \((h,k)\) is the coordinate of the vertex. By analyzing the quadratic equation, useful information can be obtained, which we explain below:

• Direction

The sign of the coefficient \(a\) specifies the direction of the quadratic function. If \(a>0\), the parabola is upward, and if \(a<0\), the parabola is downward.

• Vertex

In the vertex form, the coordinate of the vertex is \((h,k)\). In the standard form, the \(x\)-value of the vertex is equal to \(-\frac{b}{2a}\). By inserting the \(x\)-value in the equation, the \(y\)-value is obtained.

• Axis of symmetry

The axis of symmetry of the quadratic equation is the vertical line that passes through its vertex: \(x=-\frac{b}{2a}\)

• \(x\)-intercept(s)

To get the \(x\)-intercept(s), set the value of \(y\) in the equation equal to \(0\) and solve the equation. You can use the quadratic formula to get the \(x\)-intercept(s): \(x=\frac{-b±\sqrt{b^2-4ac}}{2a}\)

• \(y\)-intercept

To calculate the \(y\)-intercept, get the value of \(f(0)\).

• range

The range of the function is determined according to whether \(a\) is positive or negative and the \(x\)-value of the vertex.

• minimum/maximum value

If \(a>0\), the function will be upward and have a minimum point \(x=-\frac{b}{2a}\), and if \(a<0\) the function will be downward and have a maximum point \(x=-\frac{b}{2a}\).

Identify Characteristics of Quadratic Functions: Equations – Example 1:

Specify characteristics of the vertex, direction, and intercept for the quadratic function \(f(x)=x^2+2x+2\).

Solution:

In the standard form of a quadratic function \(f(x)=ax^2+bx+c\), the vertex is not immediately obvious. The \(x\)-coordinate for the vertex can be obtained by using the formula: \(x=-\frac{b}{2a}\),

\(x=-\frac{2}{2(1)}=-\frac{2}{2}=-1\)

Now, substitute it into the equation of function to obtain the \(y\)-coordinate:

\(y=f(-1)=(-1)^2+2(-1)+2=1-2+2=1\)

So, the vertex of the function \(f(x)=x^2+2x+2\) is the ordered pair \((-1, 1)\).

Remember that the sign of the coefficient \(x^2\) indicates the direction of the quadratic equation. Since the coefficient of \(x^2\) is \(1\), then it is upward.

To find the \(y\)-intercept of a function, evaluate the output at \(f(0)\).

\(f(0)=(0)^2+2(0)+2=2\)

Exercises for Identify Characteristics of Quadratic Functions: Equations

Specify characteristics of the vertex, direction, and intercept for the quadratic function \(f(x)=x^2-2x-3\).

vertex: \((1, -4)\)

upward

\(y\)-intercept: \((0, -3)\)

\(x\)-intercept: \((-1, 0)\) \((3, 0)\)