Question

Graph the equation: \(y=x^{2}+4\)

Short answer

The vertex is at (0, 4), and the parabola opens upwards.

Step-by-step solution

- Identify the given equation

The given equation is in the form of a quadratic function: y = x^{2} + 4

- Determine the shape and orientation of the parabola

Since the coefficient of the term with x² is positive, the parabola opens upwards. The general shape of the graph will resemble a U.

- Find the vertex of the parabola

For the equation y = x² + 4, the vertex can be found by identifying the constants in the standard form y = a(x-h)² + k. Here, h = 0 and k = 4, so the vertex is at (0, 4).

- Plot the vertex on the graph

Plot the vertex point (0,4) on the coordinate grid.

- Calculate and plot additional points

Choose a few x-values to calculate corresponding y-values with the given equation. For example, for x=-2, x=-1, x=1, and x=2: y=(-2)²+4 = 4+4 = 8 y=(-1)²+4 = 1+4 = 5 y=(1)²+4 = 1+4 = 5 y=(2)²+4 = 4+4 = 8 Plot these points: (-2,8), (-1,5), (1,5), and (2,8).

- Draw the parabola

Connect the points with a smooth, curved line to form the parabola. Make sure the curve goes through all the points and is symmetric around the y-axis.

Key concepts

Quadratic Function

A quadratic function is a type of polynomial function that is characterized by the term with a degree of two, usually written as \( y = ax^2 + bx + c \). In this standard form, \(a, b, \text{and} c\) are constants with \(a ≠ 0\). The graph of a quadratic function is a curved line called a parabola. Depending on the value of the coefficient \(a\), the parabola opens upwards or downwards:

• If \(a > 0\), the parabola opens upwards like a

Vertex

The vertex of a parabola is the highest or lowest point on its graph, depending on whether it opens downwards or upwards. For a quadratic function in the form \(y = a(x - h)^2 + k\), the vertex of the parabola is given by the point \((h, k)\). This form of the equation is called the vertex form.
The vertex can be particularly useful because it tells us a lot about the function:

• The vertex represents the maximum (if the parabola opens downwards) or minimum (if it opens upwards) value of the function.
• The x-coordinate \(h\) of the vertex also represents the axis of symmetry for the parabola.

It essentially divides the parabola into two symmetrical halves. In our exercise, the quadratic function is given in the form \(y = x^2 + 4\) and, by comparison, we can see that the vertex is at the point (0, 4). This tells us that the parabola reaches its minimum value when x = 0, and this minimum value is 4.

Parabola

A parabola is the C-shaped curve formed when graphing a quadratic function. Here are some essential features of parabolas:

• Direction: The direction in which a parabola opens is determined by the sign of the coefficient \(a\) in the quadratic function \(y = ax^2 + bx + c\). If \(a > 0\), the parabola opens upwards, resembling a 'U'. If \(a < 0\), it opens downwards.
• Vertex: As mentioned before, the vertex is the minimum or maximum point on the graph.
• Axis of Symmetry: This is a vertical line that passes through the vertex and divides the parabola into two mirror-image halves. For a function in the form \(y = a(x - h)^2 + k\), the axis of symmetry is given by the line x = h.

To graph a parabola, follow these steps:
1. Find and plot the vertex.
2. Use additional x-values to calculate corresponding y-values and plot these points.
3. Ensure that the parabola is symmetric around the y-axis and connects all the plotted points smoothly. These steps were followed in the exercise to graph \(y = x^2 + 4\) as mentioned in Step 1 through Step 6.