Problem 9
Question
Sketch the graph of the function and compare it with the graph of \(y=x^{2}\) \(y=-x^{2}\)
Step-by-Step Solution
Verified Answer
The graph of \(y = -x^{2}\) is a parabola opening downwards, whereas the graph of \(y = x^{2}\) is a parabola opening upwards. This makes the two graphs look like mirror images of each other in the x-axis.
1Step 1: Graph the function \(y = x^{2}\)
The function \(y = x^{2}\) is a basic quadratic that opens up. The vertex of the graph is at the origin and its axis of symmetry is the y-axis. As x moves away from 0 in either the positive or negative direction, \(y = x^{2}\) increases.
2Step 2: Graph the function \(y = -x^{2}\)
The function \(y = -x^{2}\) is also a quadratic, but it opens down. This is due to the negative sign in front of \(x^{2}\). Like \(y = x^{2}\), the vertex of \(y = -x^{2}\) is at the origin and its axis of symmetry is the y-axis. However as x moves away from 0 in either the positive or negative direction, \(y = -x^{2}\) decreases.
3Step 3: Compare the two graphs
Both graphs are symmetric about the y-axis and have their vertex at the origin. However, they open in opposite directions. The graph of \(y = x^{2}\) opens upwards, while the graph of \(y = -x^{2}\) opens downwards. This makes them look like mirror images of each other in the x-axis.
Key Concepts
Quadratic EquationsAxis of SymmetryVertex of a ParabolaGraph Transformations
Quadratic Equations
Understanding quadratic equations is essential when dealing with their graphical representations. A quadratic equation is traditionally written in the form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients, and \(a \eq 0\). When graphing, \(a\) determines the direction the parabola opens—positive values mean it opens upwards, while negative values indicate it opens downwards, as showcased in the exercise with \(y=x^2\) and \(y=-x^2\).
It's crucial to note how the coefficients \(b\) and \(c\) influence the position of the parabola on the graph, although they were not a part of the given exercise. Changing \(b\) shifts the parabola horizontally, and adjusting \(c\) moves it vertically without changing its shape. Recognizing these effects helps in predicting and sketching the graph accurately.
It's crucial to note how the coefficients \(b\) and \(c\) influence the position of the parabola on the graph, although they were not a part of the given exercise. Changing \(b\) shifts the parabola horizontally, and adjusting \(c\) moves it vertically without changing its shape. Recognizing these effects helps in predicting and sketching the graph accurately.
Axis of Symmetry
The axis of symmetry in a parabola is a vertical line that divides the graph into two mirror-image halves. For the standard quadratic equation \(y = ax^2 + bx + c\), this line can be found using the formula \(x = -b/(2a)\).
In our exercise, both given functions \(y = x^2\) and \(y = -x^2\) have their axis of symmetry along the y-axis, which means \(b = 0\) in both cases. This axis provides a valuable reference when plotting points and ensures symmetry. When graphing quadratics, always look to plot points on either side of this axis to guide the shape of your parabola.
In our exercise, both given functions \(y = x^2\) and \(y = -x^2\) have their axis of symmetry along the y-axis, which means \(b = 0\) in both cases. This axis provides a valuable reference when plotting points and ensures symmetry. When graphing quadratics, always look to plot points on either side of this axis to guide the shape of your parabola.
Vertex of a Parabola
The vertex of a parabola is a crucial point where the graph changes direction; it is either the highest or lowest point on the graph, depending on whether the parabola opens downwards or upwards, respectively.
For the equations in our exercise, the vertices are at the origin \( (0, 0) \) because the parabolas are in standard form without any horizontal or vertical shifts. The vertex form of a quadratic equation is \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. By understanding vertex form, students can easily identify or manipulate the vertex to graph or translate the parabola as needed.
For the equations in our exercise, the vertices are at the origin \( (0, 0) \) because the parabolas are in standard form without any horizontal or vertical shifts. The vertex form of a quadratic equation is \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. By understanding vertex form, students can easily identify or manipulate the vertex to graph or translate the parabola as needed.
Graph Transformations
Graph transformations allow us to alter a known graph, such as that of \(y=x^2\), to understand how other quadratic functions will look.
In our exercise, the transformation involves reflecting the graph of \(y=x^2\) across the x-axis to get \(y=-x^2\). This reflection changes the direction in which the parabola opens. Other transformations include translations (shifting the graph up, down, left, or right), stretches (making the parabola narrower or wider), and compressions (making it more flat). Students should emphasize mastering transformations as they provide a dynamic way to quickly grasp changes in the graph without plotting point by point.
In our exercise, the transformation involves reflecting the graph of \(y=x^2\) across the x-axis to get \(y=-x^2\). This reflection changes the direction in which the parabola opens. Other transformations include translations (shifting the graph up, down, left, or right), stretches (making the parabola narrower or wider), and compressions (making it more flat). Students should emphasize mastering transformations as they provide a dynamic way to quickly grasp changes in the graph without plotting point by point.
Other exercises in this chapter
Problem 9
Confirm that the function has the indicated zeros. $$f(x)=x^{2}+5 ;-\sqrt{5} i, \sqrt{5} i$$
View solution Problem 9
Use long division to divide and use the result to factor the dividend completely. $$\left(x^{2}+5 x+6\right) \div(x+3)$$
View solution Problem 10
(a) use a graphing utility to create a scatter plot of the data, (b) determine whether the data could be better modeled by a linear model or a quadratic model,
View solution Problem 10
(a) find the domain of the function, (b) complete each table, and (c) discuss the behavior of \(f\) near any excluded \(x\)-values. $$\begin{array}{|l|l|} \hlin
View solution