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Without graphing, what are the vertex , axis of symmetry, and transformations of the parent function? Y - |8x-3| - 3

Sagot :

Given:

Consider the given function is

[tex]y=|8x-3|-3[/tex]

To find:

The vertex , axis of symmetry, and transformations of the parent function?

Solution:

We have,

[tex]y=|8x-3|-3[/tex]

[tex]y=\left|8\left(x-\dfrac{3}{8}\right)\right|-3[/tex]

[tex]y=8\left|x-\dfrac{3}{8}\right|-3[/tex]      ...(i)

It is an absolute function.

The vertex form of an absolute function is

[tex]y=a|x-h|+k[/tex]       ...(ii)

where, a is a constant, (h,k) is vertex and x=h is axis of symmetry.

From (i) and (ii), we get

[tex]a=8,h=\dfrac{3}{8},k=-3[/tex]

So,

[tex]\text{Vertex}:(h,k)=\left(\dfrac{3}{8},-3\right)[/tex]

[tex]\text{Axis of symmetry}:x=\dfrac{3}{8}[/tex]

Parent function of an absolute function is

[tex]y=|x|[/tex]

Since, a=8 therefore, parent function vertically stretched by factor 8.

[tex]h=\dfrac{3}{8}>0[/tex], so the function shifts [tex]\dfrac{3}{8}[/tex] unit right.

k=-3<0, so the function shifts 3 units down.

Therefore, the vertex is [tex]\left(\dfrac{3}{8},-3\right)[/tex] and Axis of symmetry is [tex]x=\dfrac{3}{8}[/tex]. The parent function vertically stretched by factor 8, shifts [tex]\dfrac{3}{8}[/tex] unit right and  3 units down.