Question

Use symmetry to complete the graph of the equation. \(x\) -axis symmetry \(y=-x^{2}+4\)

Short answer

To draw the symmetric graph of the given equation, first plot the vertex at (0,4) and identify the y-axis as the axis of symmetry. Then plot additional points like (1,3), (-1,3), (2,0), and (-2,0). Finally, draw a smooth curve through these points to complete the graph.

Step-by-step solution

Identify the Vertex

The vertex of the parabola is given by the point (0, 4), since the equation is already adjusted in the form \(y=a(x-h)^{2}+k\) with h=0 and k=4.

Plot the Vertex and Axis of Symmetry

Plot the vertex point (0,4) on the graph. Then draw the axis of symmetry, which is the vertical line \(x=0\) or the y-axis.

Find and Plot additional points

Since the equation is \(y=-x^{2}+4\), pick a few values for x, and compute the corresponding y values. For example when \(x=1, y=-1^{2}+4=3\), and when \(x=-1, y=-(-1)^{2}+4=3\), so the points (1,3) and (-1,3) are on the graph. Similarly, for \(x=2, y=-2^{2}+4=0\), and when \(x=-2, y=-(-2)^{2}+4=0\), so the points (2,0) and (-2,0) are on the graph. Plot these points on the graph.

Draw the Graph

Finally, draw a smooth curve through the plotted points, including the vertex, to complete the graph of the parabola.

Key concepts

Axis of Symmetry

The axis of symmetry in a graph is a straight line that divides the graph into two mirror-image halves. When graphing parabolas, which are the graphs of quadratic functions, this concept is paramount. For a quadratic function in standard form, which looks like \( y = ax^2 + bx + c \), the axis of symmetry can be found using the formula \( x = -\frac{b}{2a} \). In the case of the given equation \( y = -x^2 + 4 \), since there is no \( bx \) term, the value of \( b \) is 0. Thus, the axis of symmetry is \( x = -\frac{0}{2\times(-1)} = 0 \), which is simply the y-axis.

Understanding the axis of symmetry helps students predict the shape of the parabola and determine where to plot additional points effectively. It's a key feature that ensures the parabola is symmetric, and it helps ensure that the graph you produce will be accurate and well-proportioned.

Vertex of a Parabola

The vertex of a parabola is the peak or the bottom point where the graph changes direction. It is a crucial point as it can represent the maximum or minimum value of the quadratic function. In vertex form, which is \( y = a(x-h)^2 + k \), the vertex is at the point \( (h, k) \).

In the problem you're working on, the equation \( y = -x^2 + 4 \) is already structured such that the \( h \) and \( k \) values are identifiable. The vertex is at (0, 4) because \( h = 0 \) and \( k = 4 \). Correctly identifying and plotting the vertex is the first step in graphing a parabola, as it serves as a reference point from which you can symmetrically plot additional points on either side of the axis of symmetry.

Quadratic Functions

Quadratic functions are mathematical expressions of the form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) cannot be zero. The graph of a quadratic function is a parabola, which can open upwards (when \( a > 0 \)) or downwards (when \( a < 0 \)). The shape and position of the parabola are determined by the values of \( a \), \( b \), and \( c \).

For instance, in our current example \( y = -x^2 + 4 \), the coefficient \( a \) is negative, which means the parabola opens downwards. In addition, since the equation lacks a \( bx \) term (that is, \( b = 0 \)), the parabola is centered on the y-axis. Understanding how the coefficients influence the graph's opening, orientation, and position helps students predict and sketch quadratic function graphs accurately.

Symmetry in Graphs

Symmetry in graphs refers to a situation where two halves of the graph are mirror images of each other. In the context of parabolas, this means that if you fold the graph along the axis of symmetry, both sides will match perfectly. This characteristic is especially useful for graphing, as it allows you to calculate points on one side of the parabola and use the axis of symmetry to reflect them on the other side.

When completing the graph for the quadratic function \( y = -x^2 + 4 \), once you've plotted a point to the right of the axis of symmetry, its mirror image can be found directly to the left at the same distance from the axis. This is illustrated in the exercise by plotting points (1,3) and (-1,3), which are equidistant from the axis of symmetry, \( x=0 \). Similarly, the pair (2,0) and (-2,0) reflect across the axis. Teaching students to utilize symmetry when graphing can simplify the task and reduce potential errors.