Question

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Answer 1

Answer:        

3)$y = x^2$, $y = 2x^2$, $y = 3x^2$

4) $y = -x^2$,$y = -5x^2$, $y = -4x^2$

5) It is shifted 1 unit up.

Step-by-step explanation:

3) Since, the general equation of parabola is,

$f(x) = a(x-h)^2+k$

If 0<a<1 then f(x) is wider than $f(x)=x^2$

If a>1 then f(x) is narrow that $f(x)=x^2$

Thus, the required sequence of equations from widest to narrow graph,

$y = x^2$, $y = 2x^2$, $y = 3x^2$

4) Again if a<0 then $f(x)=-a(x-h)^2+k$ is wider than $f(x)= -x^2$

Thus, the required sequence from widest to narrowest graph is,

$y = -x^2$,$y = -5x^2$, $y = -4x^2$

5) Since, In equation of parabola,

$f(x) = a(x-h)^2+k$

k shows the y-coordinate of the vertex of the parabola,

⇒ k shows the shifting along y-axis.

Thus, If the graph $y=4x^2$ is transformed to $y=4x^2+1$

Then we will say it is shifted 1 unit up or shifted vertically by the factor 1.

Therefore, First Option is correct.

Answer 2

Question (3): The correct option is $\boxed{\bf option (d)}$.

Question (4): The correct option is $\boxed{\bf option (c)}$.

Question (5): The correct option is $\boxed{\bf option (a)}$.

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 $f(x)$ by a factor of $c$ can be obtained by $g(x)=cf(x)$ if $c$ is greater than $1$.

The graph is stretched as we increase the value of $c$ in first case the value of $c$ is $1$ and in the second case the value of $c$ is $2$ and the third case is value of $c$ is $3$.

It means that the third case has maximum stretching that leads to a narrow graph, and similarly in second case the value of $c$ is $2$ it is also stretches but less than the third case.

Therefore, the order of quadratic functions from widest to narrowest graph is as follows:

$\boxed{(y=x^{2})>(y=2x^{2})>(y=3x^{2})}$

Figure 1 (attached in the end) represents the graph of the functions $y=x^{2},y=2x^{2},y=3x^{2}$.

Thus, the correct option is $\boxed{\bf option (d)}$.

Question (4):

Solution:

The reflection of graph of $f(x)$ about $x$-axis can be obtained by $g(x)=-f(x)$.

The graph of the function $y=-x^{2}$ is obtained when each point on the curve of the function $y=x^{2}$ is reflected across the $x$-axis.

Similarly, the graph of the function $y=-4x^{2}$ $y=-5x^{2}$ and is obtained when each point on the curve of the function $y=4x^{2}$ and $y=5x^{2}$ is reflected across the $x$-axis respectively.

From figure 2 (attached in the end) it is observed that the graph of the function $y=-x^{2}$ is the widest and the graph of the function $y=-5x^{2}$ is the narrowest.

Therefore, the correct option is $\boxed{\bf option (c)}$.

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 $y=f(x)+a$ is obtained when each point on the curve of $y=f(x)$ is shifted along the $y$-axis. If $a$ is positive then the curve shifts vertically upwards and if $a$ is negative then the curve shifts vertically downwards.

Similarly, the graph of the function $y=4x^{2}+1$ is obtained when each point on the curve of $y=4x^{2}$ shifts $1\text{ unit}$ vertically upwards.

Figure 3 (attached in the end) represents the graph of the function $y=4x^{2}$ and $y=4x^{2}+1$.

Therefore, the correct option is $\boxed{\bf option (a)}$.

Learn more:

1. Representation of graph brainly.com/question/2491745

2. Quadratic equation: 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.