Question

analyzing relationships in exercises 25 and 26, describe the x-values for which (a) f is increasing, (b) f is decreasing, (c) f is positive, and (d) f is negative. 25. graph 1 26. graph 2

Explanation:

For Exercise 25:

Step1: Identify increasing interval

A function increases when it rises as $x$ increases. This happens to the right of the vertex at $x=4$.
Interval: $x > 4$

Step2: Identify decreasing interval

A function decreases when it falls as $x$ increases. This happens to the left of the vertex at $x=4$.
Interval: $x < 4$

Step3: Identify positive $f(x)$ values

$f(x)$ is positive where the graph is above the $x$-axis. The graph crosses the $x$-axis at $x=3$ and $x=5$.
Intervals: $x < 3$ or $x > 5$

Step4: Identify negative $f(x)$ values

$f(x)$ is negative where the graph is below the $x$-axis.
Interval: $3 < x < 5$

Step1: Identify increasing interval

A function increases when it rises as $x$ increases. This happens to the right of the vertex at $x=1$.
Interval: $x > 1$

Step2: Identify decreasing interval

A function decreases when it falls as $x$ increases. This happens to the left of the vertex at $x=1$.
Interval: $x < 1$

Step3: Identify positive $f(x)$ values

$f(x)$ is positive where the graph is above the $x$-axis. The graph crosses the $x$-axis at $x=-1$ and $x=2$.
Intervals: $x < -1$ or $x > 2$

Step4: Identify negative $f(x)$ values

$f(x)$ is negative where the graph is below the $x$-axis.
Interval: $-1 < x < 2$

Answer:

(a) $x > 4$
(b) $x < 4$
(c) $x < 3$ or $x > 5$
(d) $3 < x < 5$

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For Exercise 26: