Question

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

Short answer

Graph the equation by plotting the vertex (0, -4) and additional points like (1, -3), (-1, -3), (2, 0), and (-2, 0), then draw a smooth upward parabola.

Step-by-step solution

Rewrite the Equation

First, rewrite the given equation in the standard form of a parabola. The given equation is enspace y + 4 = x^2Subtract 4 from both sides to get enspace y = x^2 - 4.

Identify the Vertex

The equation is now in the form enspace y = x^2 - 4,enspace which is a parabola that opens upwards. The vertex of the parabola is at the point enspace (0, -4).

Plot the Vertex

Plot the vertex (0, -4) on a coordinate plane. This point is where the parabola will change direction.

Plot Additional Points

Choose some values for enspace x,enspace and find the corresponding enspace yenspace values using the equation enspace y = x^2 - 4.enspace For example:If enx=1,enspace then enspace y = 1^2 - 4 = -3.If x=-1,enspace thenenspace y = (-1)^2 - 4 = -3.If x=2,enspace thenenspace y = 2^2 - 4 = 0.If x=-2,enspace then y = (-2)^2 - 4 = 0.

Plot the Points

Plot the points (1, -3), (-1, -3), (2, 0), and (-2, 0) on the coordinate plane.

Draw the Parabola

Draw a smooth curve through all the plotted points to form the parabola. It should be symmetrical about the y-axis and open upwards.

Key concepts

vertex of a parabola

The vertex of a parabola is a crucial point where the direction of the curve changes. It is often referred to as the 'turning point' of the parabola. For the equation given in the exercise, we have y = x^2 - 4. The vertex can be found by examining the formula. Here, the vertex form of a parabola is written as y = (x - h)^2 + k. Since there are no shifts horizontally or vertically to 'x' (in the form of h and k respectively), the vertex is at (0, -4).

To find the vertex:

• Identify the values of h and k in the formula
• For y = x^2 - 4, h = 0 and k = -4
• The vertex is at the point (0, -4)

standard form of a parabola

The standard form of a parabola's equation is essential for understanding its properties and graphing it accurately. The standard form allows us to quickly identify the parabola's vertex and direction of opening. For a vertically oriented parabola, the standard form is given by y = ax^2 + bx + c. In this exercise, we start with y + 4 = x^2. To convert this to the standard form:

• Subtract 4 from both sides to isolate y: y = x^2 - 4
• Now we have it in the classic standard form: y = ax^2 + bx + c, where a = 1, b = 0, and c = -4

Knowing the standard form, we can easily locate the vertex and understand the parabola's behavior.

plotting points on a graph

Plotting points on a graph is a fundamental step in graphing a parabola. It involves calculating the y-values for selected x-values and then marking these points on the coordinate plane. From our equation y = x^2 - 4, let's find several key points:

• When x = 0, y = 0^2 - 4 = -4
• When x = 1, y = 1^2 - 4 = -3
• When x = -1, y = (-1)^2 - 4 = -3
• When x = 2, y = 2^2 - 4 = 0
• When x = -2, y = (-2)^2 - 4 = 0

These points are: (0, -4), (1, -3), (-1, -3), (2, 0), and (-2, 0). Plot these points on the graph to visualize the parabola.

symmetry in parabolas

Symmetry in parabolas is one of their prominent features which makes them easier to graph. A parabola that opens upwards or downwards is symmetric about its vertical line through the vertex. In the case of y = x^2 - 4, our vertex is at (0, -4), and the axis of symmetry is the y-axis itself (x = 0).

This means:

• The points on the parabola are mirrored across this axis
• If (1, -3) is a point on the parabola, then its mirror image (-1, -3) is also a point on the parabola

Using this symmetry helps in accurately graphing and visualizing the parabola.