Which of the following graphs in the $x y$ -plane have $-3$ and $5$ as all of their distinct zeros for $-6 \leq x \leq 6$ ?

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Q. Which of the following graphs in the

x y

-plane have

-3

and

5

as all of their distinct zeros for

-6 \leq x \leq 6

Problem Understanding: Understand the problem. $\newline$ We need to find the graphs that have $-3$ and $5$ as their only zeros within the interval $-6 \leq x \leq 6$ . This means that the graph should touch or cross the x-axis at $x = -3$ and $x = 5$ , and not at any other point within the given interval.
General Form of the Polynomial: Determine the general form of the polynomial. $\newline$ A polynomial that has $-3$ and $5$ as zeros can be written as $f(x) = a(x + 3)(x - 5)$ , where $a$ is a non-zero constant. The value of $a$ will affect the width and direction of the graph but not the zeros.
Additional Requirements: Check for additional requirements. $\newline$ Since the problem only specifies that $-3$ and $5$ are the distinct zeros and does not mention any other conditions, any graph of the form $f(x) = a(x + 3)(x - 5)$ with $a \neq 0$ will satisfy the condition as long as it does not cross the x-axis at any other point within the interval $-6 \leq x \leq 6$ .
Behavior at the Zeros: Determine the behavior of the graph at the zeros. $\newline$ At $x = -3$ , the graph should either touch or cross the x-axis, and the same goes for $x = 5$ . If $a > 0$ , the graph will open upwards, and if $a < 0$ , the graph will open downwards.
Behavior Outside the Zeros: Verify the behavior of the graph outside the zeros. For $x < -3$ , the graph should be above the x-axis if $a > 0$ and below the x-axis if $a < 0$ . For $-3 < x < 5$ , the graph should be below the x-axis if $a > 0$ and above the x-axis if $a < 0$ . For $x > 5$ , the graph should be above the x-axis if $a > 0$ and below the x-axis if $a < 0$ .
Graph Characteristics: Conclude the characteristics of the graph. $\newline$ The graph of $f(x) = a(x + 3)(x - 5)$ will have $-3$ and $5$ as its zeros, and it will not have any other zeros within the interval $-6 \leq x \leq 6$ . The graph will either open upwards or downwards depending on the sign of $a$ , but this does not affect the location of the zeros.