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Chapter 2: Problem 14

Sketch the graph of each quadratic function and compare it with the graph of \(y=x^{2}\). (a) \(f(x)=x^{2}+1\) (b) \(g(x)=x^{2}-1\) (c) \(h(x)=x^{2}+3\) (d) \(k(x)=x^{2}-3\)

Short Answer

Expert verified
All the given functions are translations of the graph \(y=x^{2}\): \(f(x)=x^{2}+1\) is shifted 1 unit up; \(g(x)=x^{2}-1\) shifted down by 1 unit; \(h(x)=x^{2}+3\) shifted 3 units up; \(k(x)=x^{2}-3\) shifted 3 units down.

Step by step solution

01

Consider the original function

Begin by examining the function \(y=x^{2}\). Its graph is a parabola that opens upward, with vertex at the origin (0,0). This will be the baseline comparison for the other functions.
02

Graph \(f(x)=x^{2}+1\)

The graph of \(f(x)=x^{2}+1\) is same as \(y=x^{2}\), but shifted upward 1 unit because adding a constant to a function shifts the graph of that function upward by that constant. Thus, its vertex is at (0,1).
03

Graph \(g(x)=x^{2}-1\)

The graph of \(g(x)=x^{2}-1\) is the same as \(y=x^{2}\), but shifted downward by 1 unit because subtracting a constant from a function shifts the graph of that function downward by that constant. So, its vertex is at (0,-1).
04

Graph \(h(x)=x^{2}+3\)

In the case of \(h(x)=x^{2}+3\), the graph of \(y=x^{2}\) is shifted upward by 3 units. The vertex is at (0,3).
05

Graph \(k(x)=x^{2}-3\)

The function \(k(x)=x^{2}-3\) represents a parabola that is shifted downward by 3 units from \(y=x^{2}\), hence the vertex is at (0,-3).

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Most popular questions from this chapter

Write the polynomial as the product of linear factors and list all the zeros of the function. $$h(x)=x^{2}-2 x+17$$

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Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. \(f(x)=x^{4}-4 x^{3}+5 x^{2}-2 x-6\) (Hint: One factor is \(\left.x^{2}-2 x-2 .\right)\)

Match the cubic function with the numbers of rational and irrational zeros. (a) Rational zeros: \(0 ;\) irrational zeros: 1 (b) Rational zeros: \(3 ;\) irrational zeros: 0 (c) Rational zeros: \(1 ;\) irrational zeros: 2 (d) Rational zeros: \(1 ;\) irrational zeros: 0 $$f(x)=x^{3}-2 x$$
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