Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Nov 03, 2023

Écrivez une fonction polynomiale satisfaisant les conditions: a) $x$ dans les quadrants III-IV, 1 point d'inflexion, $y$ -int: 2. b) $x$ dans les quadrants III-I, 3 abscisses à l'origine. c) croissante, degré 1, $y$ -int: -3. d) 2 points d'inflexion, $y$ -int: 5. e) $y \leq 2$ , $y$ -int: 2.

STEP 1

Assumptions1. We are asked to write the equation of a polynomial function that satisfies each set of characteristics. . We are assuming that the polynomial function is in the form $f(x) = ax^n + bx^{n-1} + ... + k$ where $a, b, ..., k$ are constants, $x$ is the variable, and $n$ is the degree of the polynomial.
3. The characteristics given in each part of the question provide information about the shape, degree, and certain points of the polynomial function.

STEP 2

For part a), we are given that the function extends from quadrant III to quadrant IV, has a turning point, and the y-intercept is2. This suggests a cubic function of the form $f(x) = ax^ + bx^2 + cx + d$ where $a<0$ (since the function extends from quadrant III to quadrant IV), and $d=2$ (since the y-intercept is2).
$f(x) = ax^ + bx^2 + cx +2$

STEP 3

For part b), we are given that the function extends from quadrant III to quadrant I, and has three x-intercepts. This suggests a cubic function of the form $f(x) = ax^3 + bx^2 + cx + d$ where $a>0$ (since the function extends from quadrant III to quadrant I), and the roots of the function are the x-intercepts. Let's say the x-intercepts are $r, s, t$ . Then the function can be written as $f(x) = a(x-r)(x-s)(x-t)$

STEP 4

For part c), we are given that the function is increasing, of degree1, and the y-intercept is -3. This suggests a linear function of the form $f(x) = ax + b$ where $a>0$ (since the function is increasing), and $b=-3$ (since the y-intercept is -3).
$f(x) = ax -3$

STEP 5

For part d), we are given that the function has two turning points, and the y-intercept is5. This suggests a quartic function of the form $f(x) = ax^4 + bx^3 + cx^2 + dx + e$ where $e=5$ (since the y-intercept is5).
$f(x) = ax^4 + bx^3 + cx^2 + dx +5$

STEP 6

For part e), we are given that the range of the function is $y \leq2$ , and the y-intercept is2. This suggests a constant function of the form $f(x) = k$ where $k=2$ (since the y-intercept is2 and the range is $y \leq2$ ).
$f(x) =2$

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