Problem 38 - College Algebra

Question

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $$f(x)-6-4 x+x^{2}$$

Step-by-Step Solution

Answer

The vertex of the quadratic function \(f(x)=x^2-4x+6\) is at the point (2,2). The equation of the axis of symmetry is \(x=2\). The domain of the function is \(-\infty < x < \infty\), and the range is \(y \geq 2\).

Step 1: Find the Vertex

The vertex coordinates of a quadratic equation \(f(x)=ax^2+bx+c\) are given by \((-b/2a , f(-b/2a))\). For the given equation \(f(x)=x^2-4x+6\), \(a=1\), \(b=-4\) and \(c=6\). Therefore vertex is \((-b/2a , f(-b/2a)) = (2, 2).\)

Step 2: Determine the Axis of Symmetry

The axis of symmetry of a parabola is a vertical line passing through the vertex of the parabola. The x-coordinate of the vertex gives the equation of axis of symmetry, which is \(x=-b/2a=2\).

Step 3: Define the Range and Domain of the Function

The domain for any quadratic function is all real numbers because \(x\) can take any real value. Hence, Domain: \(-\infty < x < \infty\). The range is all possible values of \(y\) that we get after substituting all the domain (x) values in the function. Since the given quadratic function opens upwards and the minimum point (vertex) is (2, 2), Range: \(y \geq 2\)

Step 4: Sketch the Graph

Plot the vertex point (2,2) and the line of symmetry \(x=2\) on the graph, and sketch the graph of the function. The graph opens up because \(a>0\), the vertex (2,2) is the lowest point and all \(y\) values are equal to or greater than 2.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial point that represents either its highest or lowest point. In the context of quadratic functions denoted by the equation \( f(x) = ax^2 + bx + c \), the vertex can be found using the formula \( (-b/2a, f(-b/2a)) \). This gives both the x-coordinate and the y-coordinate of the vertex. For the quadratic equation given in the exercise, \( f(x) = x^2 - 4x + 6 \), we calculate the vertex as follows:

• First, find \( x = -b/2a \). With \( a = 1 \) and \( b = -4 \), we find \( x = 2 \).
• Next, substitute \( x \) back into the function to find \( y \). This results in \( y = f(2) = 2 \).

Thus, the vertex is at the point \( (2, 2) \). This point tells us that the parabola has a minimum point at \( y = 2 \) and that it opens upwards since \( a \) is positive.

Axis of Symmetry

The axis of symmetry for a parabola is a vertical line that passes through the vertex, and it divides the parabola into two mirror-image halves. For a quadratic function \( f(x) = ax^2 + bx + c \), the axis of symmetry is found at the x-coordinate of the vertex, which is given by \( x = -b/2a \). It's essential for understanding the parabolic shape because it helps to predict points on the graph:

• For our function \( f(x) = x^2 - 4x + 6 \), the axis of symmetry is \( x = 2 \).

By knowing the axis of symmetry, you can reflect points across this line to find additional points on the graph, ensuring you keep an accurate, symmetrical shape. It acts as a guide for sketching and analyzing the parabola.

Domain and Range

In mathematics, the domain and range of a function are the sets of inputs and outputs, respectively. For quadratic functions, the domain is

• All real numbers, represented as \(-\infty < x < \infty\)

since there are no restrictions or exclusions on x-values in standard quadratic functions.

The range of a quadratic function, however, is influenced by the direction it opens and the vertex. As the parabola described in the exercise opens upwards, and its vertex is \( (2, 2) \), the lowest possible value of \( y \) is 2.

• Thus, the range is given by \( y \geq 2 \).

Knowing the domain and range helps to fully describe and understand the behavior and limitations of quadratic functions.

Graph of a Quadratic Function

Visualizing the graph of a quadratic function can often offer key insights into its behavior and characteristics. When graphing a quadratic function like \( f(x) = x^2 - 4x + 6 \), start with:

• Plot the vertex point \((2, 2)\) on the coordinate plane.
• Mark the axis of symmetry \( x = 2 \), which helps ensure that your drawing is precise and symmetrical.

Because the quadratic opens upwards (since \( a = 1 \) is positive), the curve of the parabola will rise from the vertex. You can then add additional points using symmetry or by evaluating the function at various x-values.

The graph visually represents all possible solutions \((x, y)\), shows the minimum point at the vertex, and confirms that the range of the function starts from \( y = 2 \) and goes upwards to infinity. Sketching this curve is a practical way to amalgamate the pure algebraic understanding with geometry.