Problem 11 - Algebra 1: Concepts and Skills

Question

Sketch the graph of the function. Label the coordinates of the vertex. Write an equation for the axis of symmetry. $$ y=x^{2}+4 $$

Step-by-Step Solution

Answer

The vertex of the function y=x^2+4 is at (0, 4). The graph is a parabola that opens upwards. The equation of the axis of symmetry is x = 0.

Step 1: Identify the Vertex of the Function

Since the given quadratic function \(y = x^2 + 4\) is in the standard form \(y = x^2 + k\), where \(k = 4\), there’s no \((x-h)\) in the function therefore \(h = 0\). The vertex will be at the point \((h, k)\), or \((0, 4)\).

Step 2: Sketch the Graph

Draw x and y axes on a piece of paper. Label the y-axis at the point (0, 4) as the vertex. Since the coefficient of \(x^2\) in the function is positive, the parabola opens upwards. Plot a couple of additional points on either side of the vertex such as (-2, 8) and (2, 8) and draw a smooth curve through the points.

Step 3: Determine the Equation for the Axis of Symmetry

The equation for the axis of symmetry is \(x = h\), and since \(h = 0\), the equation is \(x = 0\).

Key Concepts

Vertex Form

The vertex form of a quadratic function is often expressed as \(y = a(x-h)^2 + k\). This form is particularly useful because it clearly shows the vertex of the parabola, which is the point \((h, k)\). For the function \(y = x^2 + 4\), although it’s not explicitly in vertex form, we can compare it to the simpler version \(y = x^2 + k\). Here, the absence of \((x-h)\) implies \(h = 0\), and \(k = 4\), which easily identifies the vertex as \((0, 4)\). The vertex is a crucial point on the graph of a quadratic function. It represents the minimum or maximum point of the parabola. In this example, since the coefficient of \(x^2\) is positive, the parabola opens upwards, making the vertex \((0, 4)\) the lowest point on the graph.

Axis of Symmetry

The axis of symmetry for a parabola is a vertical line that passes through the vertex, dividing the parabola into two mirrored halves. It can be understood as a kind of 'mirror line' for the graph. For any quadratic function in the form \(y = a(x-h)^2 + k\) or \(y = x^2 + k\), the axis of symmetry can be found using the equation \(x = h\). In the given function \(y = x^2 + 4\), since we identified the vertex \(h = 0\), the equation for the axis of symmetry is \(x = 0\). This line goes through the vertex at \((0, 4)\) and shows where the left and right sides of the parabola are symmetric. This concept helps in efficiently plotting the graph as it indicates that points equidistant from the axis will have the same \(y\)-value.

Parabola Graphing

Graphing a parabola involves plotting its vertex and using the axis of symmetry to aid in sketching the curve. Begin by plotting the vertex point, which for \(y = x^2 + 4\), is \((0, 4)\). This is the key point from which the parabola opens upwards. Since the axis of symmetry is \(x = 0\), the parabola is mirrored on this line.

• First, plot the point \((0, 4)\) on a coordinate grid to mark the vertex.
• Next, select additional points either on the left or right of the vertex; for instance, \((-2, 8)\) and \((2, 8)\).
• Map these points accurately, ensuring each has a corresponding point across the axis of symmetry.

Finally, connect these points with a smooth curve to form the parabola. Make sure the curve passes through all identified points and mirrors evenly about \(x = 0\). Graphing helps visualize the function’s behavior and how rapidly or slowly the values change as \(x\) moves away from the vertex.