Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 3 - Review - Concept Check - Page 319: 1

Answer

$(a)$ A quadratic function has a form $ax^2+bx+c$, it has highest degree of $2$. So it's just a polynomial of degree $2$. Quadratic function has a standard form of $f(x)=a(x-h)^2+k$ To put a quadratic function into standard form, we have to simply complete the square. $(b)$ The vertex of the quadratic function that has a standard form is $(h,k)$ The vertex of a parabola is either it's maximum or minimum value, that is determined by the sign of first coefficient $a$. If the coefficient is negative, then the graph of the function opens downward and it has maximum value. If the coefficient is positive, then the graph of the function opens upward and it has minimum value. $(c)$ $f(x)=(x+2)^2-3$ The parabola opens upward and it has vertex $(-2,-3)$ The minimum value is $f(-2)=3$

Work Step by Step

$(a)$ A quadratic function has a form $ax^2+bx+c$, it has highest degree of $2$. So it's just a polynomial of degree $2$. Quadratic function has a standard form of $f(x)=a(x-h)^2+k$ To put a quadratic function into standard form, we have to simply complete the square. $(b)$ The vertex of the quadratic function that has a standard form is $(h,k)$ The vertex of a parabola is either it's maximum or minimum value, that is determined by the sign of first coefficient $a$. If the coefficient is negative, then the graph of the function opens downward and it has maximum value. If the coefficient is positive, then the graph of the function opens upward and it has minimum value. $(c)$ To get the standard form we have to complete the square, we simply add and subtract $4$ : $f(x)=(x^2+4x\underline{+4})+1\underline{-4}$ $f(x)=(x+2)^2-3$ The parabola opens upward and it has vertex $(-2,-3)$ The minimum value is $f(-2)=3$
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