We are asked to solve the inequality $x^2 - 5x + 6 < 0$. Also, given a graph, we need to determine (a) the range of the function, (b) the zeros of the function, (c) the y-intercept, (d) the equation of the axis of symmetry, and (e) the variation of the function.

1. Problem Description

We are asked to solve the inequality $x^2 - 5x + 6 < 0$. Also, given a graph, we need to determine (a) the range of the function, (b) the zeros of the function, (c) the y-intercept, (d) the equation of the axis of symmetry, and (e) the variation of the function.

2. Solution Steps

First, let's solve the inequality $x^2 - 5x + 6 < 0$.

We can factor the quadratic expression:

$x^2 - 5x + 6 = (x-2)(x-3)$.

So we want to find the values of $x$ for which $(x-2)(x-3) < 0$.

We can analyze the sign of the expression by considering the intervals determined by the roots $x=2$ and $x=3$.

If $x < 2$, then $x-2 < 0$ and $x-3 < 0$, so $(x-2)(x-3) > 0$.

If $2 < x < 3$, then $x-2 > 0$ and $x-3 < 0$, so $(x-2)(x-3) < 0$.

If $x > 3$, then $x-2 > 0$ and $x-3 > 0$, so $(x-2)(x-3) > 0$.

Therefore, the solution to the inequality $x^2 - 5x + 6 < 0$ is $2 < x < 3$.

Now let's analyze the graph:

a) The range (contradomínio) of the function is the set of all possible $y$-values. From the graph, the maximum $y$-value is approximately $-3.25$. The graph extends downwards indefinitely. So the range is $(-\infty, -3.25]$.

b) The zeros of the function are the $x$-values where the graph intersects the $x$-axis. From the graph, the zeros are approximately $x = 0.8$ and $x = 4.2$.

c) The y-intercept (ordenada na origem) is the $y$-value where the graph intersects the $y$-axis. From the graph, the y-intercept is approximately $y = -4$.

d) The equation of the axis of symmetry is a vertical line that passes through the vertex of the parabola. The vertex appears to be at $x = 2.5$. So the equation of the axis of symmetry is $x = 2.5$.

e) The variation of the function describes how the $y$-value changes as $x$ increases.

The function is increasing for $x < 2.5$ and decreasing for $x > 2.5$.

3. Final Answer

Solution to the inequality: $2 < x < 3$

Analysis of the graph:

a) Range: $(-\infty, -3.25]$

b) Zeros: $x \approx 0.8$ and $x \approx 4.2$

c) Y-intercept: $y \approx -4$

d) Axis of symmetry: $x = 2.5$

e) Increasing for $x < 2.5$, decreasing for $x > 2.5$