Problem 4 - Calculus with Concepts in Calculus

Question

Sketch the graph of the function. $$ f(x)=x^{2}+2 $$

Step-by-Step Solution

Answer

The graph is a vertical parabola opening upwards with vertex at \((0, 2)\).

Step 1: Identify the Function Type

The given function is \( f(x) = x^2 + 2 \), which is a quadratic function. This is characterized by its parabolic shape, with the general form \( ax^2 + bx + c \). Here, \( a = 1 \), \( b = 0 \), and \( c = 2 \). The parabola opens upwards because \( a > 0 \).

Step 2: Determine the Vertex of the Parabola

The vertex form of a quadratic function is \( f(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex. For the function \( f(x) = x^2 + 2 \), \( h = 0 \) and \( k = 2 \) because it can be rewritten as \( f(x) = (x-0)^2 + 2 \). Thus, the vertex is \((0, 2)\).

Step 3: Identify the Axis of Symmetry

For any quadratic function, the axis of symmetry can be found using \( x = h \). Since \( h = 0 \), the axis of symmetry for \( f(x) = x^2 + 2 \) is the vertical line \( x = 0 \).

Step 4: Calculate Additional Points

To sketch the parabola, calculate additional points on either side of the vertex. For example, substitute \( x = 1 \) and \( x = -1 \) into \( f(x) \) to get \( f(1) = 1^2 + 2 = 3 \) and \( f(-1) = (-1)^2 + 2 = 3 \). These points \((1, 3)\) and \((-1, 3)\) help define the shape.

Step 5: Sketch the Graph

Plot the vertex \((0, 2)\) and the points \((1, 3)\) and \((-1, 3)\) on the coordinate plane. Draw a smooth curve through these points, forming a parabola that opens upward, with the vertex as the minimum point and the axis of symmetry along \( x = 0 \).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a fundamental aspect when dealing with quadratic functions. It is the point where the parabola changes direction and can be either a maximum or minimum point based on the orientation of the parabola. Quadratic functions are logically presented in the vertex form as \( f(x) = a(x-h)^2 + k \), where \((h, k)\) directly signals the vertex.

For the function \( f(x) = x^2 + 2 \), we can express it as \( f(x) = (x-0)^2 + 2 \). This makes it clear that the vertex \( (h, k) \) is positioned at \( (0, 2) \). It signifies

• the highest or lowest point on the graph,
• the minimum point for upward-facing parabolas like this one,
• the starting point for graphing.

Understanding the vertex helps in graphing and analyzing the function's behavior.

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. For quadratics in vertex form, the axis of symmetry is always expressed by \( x = h \).

In the function \( f(x) = x^2 + 2 \), since \( h = 0 \), the axis of symmetry is the line \( x = 0 \). This means:

• the parabola is symmetric with respect to the y-axis,
• each point on one side has a matching point on the other side with equal distance to the axis,
• it is a crucial feature for graph plotting and analyzing symmetry.

Recognizing this symmetry simplifies the graphing process as it guides the placement of additional points.

Additional Points on Graph

While the vertex is a key starting point, additional points around it help to sketch a more accurate shape of the parabola. By selecting x-values around the vertex and finding their corresponding y-values, we can outline the curvature properly.

For our specific function \( f(x) = x^2 + 2 \), let's look at points like \( x = 1 \) and \( x = -1 \):

• \( f(1) = 1^2 + 2 = 3 \), giving the point \( (1, 3) \),
• \( f(-1) = (-1)^2 + 2 = 3 \), giving the point \( (-1, 3) \).

These two points, along with the vertex, create a basic frame for drawing the parabola. By plotting these points and connecting them smoothly, the full upward opening parabola is formed.