Answer:
3)[tex]y = x^2[/tex], [tex]y = 2x^2[/tex], [tex]y = 3x^2[/tex]
4) [tex]y = -x^2[/tex],[tex]y = -5x^2[/tex], [tex]y = -4x^2[/tex]
5) It is shifted 1 unit up.
Step-by-step explanation:
3) Since, the general equation of parabola is,
[tex]f(x) = a(x-h)^2+k[/tex]
If 0<a<1 then f(x) is wider than [tex]f(x)=x^2[/tex]
If a>1 then f(x) is narrow that [tex]f(x)=x^2[/tex]
Thus, the required sequence of equations from widest to narrow graph,
[tex]y = x^2[/tex], [tex]y = 2x^2[/tex], [tex]y = 3x^2[/tex]
4) Again if a<0 then [tex]f(x)=-a(x-h)^2+k[/tex] is wider than [tex]f(x)= -x^2[/tex]
Thus, the required sequence from widest to narrowest graph is,
[tex]y = -x^2[/tex],[tex]y = -5x^2[/tex], [tex]y = -4x^2[/tex]
5) Since, In equation of parabola,
[tex]f(x) = a(x-h)^2+k[/tex]
k shows the y-coordinate of the vertex of the parabola,
⇒ k shows the shifting along y-axis.
Thus, If the graph [tex]y=4x^2[/tex] is transformed to [tex]y=4x^2+1[/tex]
Then we will say it is shifted 1 unit up or shifted vertically by the factor 1.
Therefore, First Option is correct.
Question (3): The correct option is [tex]\boxed{\bf option (d)}[/tex].
Question (4): The correct option is [tex]\boxed{\bf option (c)}[/tex].
Question (5): The correct option is [tex]\boxed{\bf option (a)}[/tex].
Further explanation:
Question (3):
Solution:
The best method to find which graph is widest and narrowest is graphing a quadratic function.
The vertically stretching of graph of [tex]f(x)[/tex] by a factor of [tex]c[/tex] can be obtained by [tex]g(x)=cf(x)[/tex] if [tex]c[/tex] is greater than [tex]1[/tex].
The graph is stretched as we increase the value of [tex]c[/tex] in first case the value of [tex]c[/tex] is [tex]1[/tex] and in the second case the value of [tex]c[/tex] is [tex]2[/tex] and the third case is value of [tex]c[/tex] is [tex]3[/tex].
It means that the third case has maximum stretching that leads to a narrow graph, and similarly in second case the value of [tex]c[/tex] is [tex]2[/tex] it is also stretches but less than the third case.
Therefore, the order of quadratic functions from widest to narrowest graph is as follows:
[tex]\boxed{(y=x^{2})>(y=2x^{2})>(y=3x^{2})}[/tex]
Figure 1 (attached in the end) represents the graph of the functions [tex]y=x^{2},y=2x^{2},y=3x^{2}[/tex].
Thus, the correct option is [tex]\boxed{\bf option (d)}[/tex].
Question (4):
Solution:
The reflection of graph of [tex]f(x)[/tex] about [tex]x[/tex]-axis can be obtained by [tex]g(x)=-f(x)[/tex].
The graph of the function [tex]y=-x^{2}[/tex] is obtained when each point on the curve of the function [tex]y=x^{2}[/tex] is reflected across the [tex]x[/tex]-axis.
Similarly, the graph of the function [tex]y=-4x^{2}[/tex] [tex]y=-5x^{2}[/tex] and is obtained when each point on the curve of the function [tex]y=4x^{2}[/tex] and [tex]y=5x^{2}[/tex] is reflected across the [tex]x[/tex]-axis respectively.
From figure 2 (attached in the end) it is observed that the graph of the function [tex]y=-x^{2}[/tex] is the widest and the graph of the function [tex]y=-5x^{2}[/tex] is the narrowest.
Therefore, the correct option is [tex]\boxed{\bf option (c)}[/tex].
Question (5):
Solution:
If a constant is added to a function, the graph of the function shifts vertically upwards if the constant is positive and it shifts vertically downwards if the constant is negative.
For example: The graph of the function of the form [tex]y=f(x)+a[/tex] is obtained when each point on the curve of [tex]y=f(x)[/tex] is shifted along the [tex]y[/tex]-axis. If [tex]a[/tex] is positive then the curve shifts vertically upwards and if [tex]a[/tex] is negative then the curve shifts vertically downwards.
Similarly, the graph of the function [tex]y=4x^{2}+1[/tex] is obtained when each point on the curve of [tex]y=4x^{2}[/tex] shifts [tex]1\text{ unit}[/tex] vertically upwards.
Figure 3 (attached in the end) represents the graph of the function [tex]y=4x^{2}[/tex] and [tex]y=4x^{2}+1[/tex].
Therefore, the correct option is [tex]\boxed{\bf option (a)}[/tex].
Learn more:
1. Representation of graph https://brainly.com/question/2491745
2. Quadratic equation: https://brainly.com/question/1332667
Answer details:
Grade: High school
Subject: Mathematics
Topic: Shifting of graph
Keywords: Graph,inequality ,y=x^2 ,y=-4x^2 ,y=-5x^2 ,y=x^2 ,y=2x^2 , y=3x^2, shifted, stretches, widest, narrowest, quadratic function shifting, translation, curve.
