Sovi AI

問題

解法ステップ

1. Understand the question

   use end behavior to select the correct function for each graph:
   a. $x^2 - 7x + 9$
   b. $-x^4 + 2x^2$
   c. $-x^3 - x^2 + 2x - 3$
   d. $x^3 - 4x^2$

2. Explanation

   Step1: Identify degree/leading coefficient rules

   For polynomial $f(x)=a_nx^n+...+a_0$:

   • Even $n$: $\lim_{x\to\pm\infty}f(x)=+\infty$ if $a_n>0$; $\lim_{x\to\pm\infty}f(x)=-\infty$ if $a_n<0$
   • Odd $n$: $\lim_{x\to+\infty}f(x)=+\infty, \lim_{x\to-\infty}f(x)=-\infty$ if $a_n>0$; $\lim_{x\to+\infty}f(x)=-\infty, \lim_{x\to-\infty}f(x)=+\infty$ if $a_n<0$

   Step2: Analyze top (green) graph

   End behavior: $x\to+\infty, f(x)\to+\infty$; $x\to-\infty, f(x)\to-\infty$. This matches odd degree, positive leading coefficient.
   Check options: d. $x^3-4x^2$ (degree 3, $a_3=1>0$)

   Step3: Analyze second (red) graph

   End behavior: $x\to\pm\infty, f(x)\to-\infty$. This matches even degree, negative leading coefficient.
   Check options: b. $-x^4+2x^2$ (degree 4, $a_4=-1<0$)

   Step4: Analyze third (orange) graph

   End behavior: $x\to\pm\infty, f(x)\to+\infty$. This matches even degree, positive leading coefficient.
   Check options: a. $x^2-7x+9$ (degree 2, $a_2=1>0$)

   Step5: Analyze bottom (blue) graph

   End behavior: $x\to+\infty, f(x)\to-\infty$; $x\to-\infty, f(x)\to+\infty$. This matches odd degree, negative leading coefficient.
   Check options: c. $-x^3-x^2+2x-3$ (degree 3, $a_3=-1<0$)

3. Final answer

   Top (green) graph: d. $x^3 - 4x^2$
   Second (red) graph: b. $-x^4 + 2x^2$
   Third (orange) graph: a. $x^2 - 7x + 9$
   Bottom (blue) graph: c. $-x^3 - x^2 + 2x - 3$

答え

Question Analysis

OCR Text

use end behavior to select the correct function for each graph:
a. $x^2 - 7x + 9$
b. $-x^4 + 2x^2$
c. $-x^3 - x^2 + 2x - 3$
d. $x^3 - 4x^2$

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