How is the graph of #g(x)=0.25x^2-1.1# related to the graph of #f(x)=x^2#?

Answer 1

Please read the explanation.

For a parabola #ax^2+bx+c#,

the x-coordinate value of the Vertex is given by

#-b/(2a)#

If #a<0#, then the vertex is a maximum value.

If #a>0#, then the vertex is a minimum value.

#color(blue)("Step 1"#

We have

#y=f(x)=x^2#

Hence, #a=1; b=0 and c=0#

x-coordinate of the vertex is #-b/(2a#

#rArr -0/(2*(1)#

#rArr x=0#

To obtain the y-coordinate value of the vertex,

substitute #x=0# in #y=f(x)=x^2#

#y=(0)^2#

#rArr y = 0#

Hence, the vertex of #y=f(x)=x^2# is at #(0,0)#

Since, #a>0#, vertex is a minimum.

#color(blue)("Step 2"#

Next, consider

#y=g(x)=0.25x^2-1.1#

#a=0.25; b=0 and c=-1.1#

x-coordinate of the vertex is #-b/(2a#

#rArr = -0/(2(0.25)#

#x=0#

To obtain the y-coordinate value of the vertex,

substitute #x=0# in #y=g(x)=0.25x^2-1.1#

#y=(0.25)(0^2)-1.1#

#y=-1.1#

Hence, the vertex of #y=g(x)=0.25x^2-1.1# is at #(0,-1.1)#

Since, #a>0#, vertex is a minimum.

#color(blue)("Step 3"#

Next, we will examine the graphs of both #f(x)=x^2# and #g(x)=0.25x^2-1.1#, and compare them.

#y=f(x)=x^2# is the Parent function.

The function #y=f(x)=x^2+b# has a graph with vertex shifted #b# units along the y-axis.

Then the vertex is located at #(0,b)#.

If #b# is positive, then the parabola is translated upwards and, if #b# is negative, it moves downwards.

The parabola, #y=f(x)=ax^2#, is obtained from the parent function #y= f(x)=x^2# by stretching it in the y-direction, away from the x-axis, by a factor of #a#.

Vertex is at #(0,-1,1)#

#color(blue)("Step 4"#

For #g(x)=0.25x^2-1.1#, the arms of the parabola are steeper than those of the parent function.

The y-values of each point on this parabola is #(0.25)# times the y-value of the point on the parent function with the same x-value.

Hence, the graph has been stretched in the y-direction by a factor of #0.25#.

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Answer 2

The graph of ( g(x) = 0.25x^2 - 1.1 ) is related to the graph of ( f(x) = x^2 ) by a vertical translation downward by 1.1 units and a horizontal compression by a factor of 0.25.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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