Question

use the graph to estimate the x- and y-intercepts of the function and describe where the function is positive and negative.
x-intercept(s): select choice
y-intercept: select choice
positive: when select choice
negative: when select choice

Explanation:

Step1: Identify x - intercepts

The x - intercepts are the points where the graph crosses the x - axis (where \(y = 0\)). From the graph, we can see that the graph crosses the x - axis at \(x=1\) and \(x = - 1\) (assuming the grid has a scale where each square is 1 unit). Wait, looking at the graph again, let's check the coordinates. Let's assume the origin \(O\) is at \((0,0)\). The left - hand part of the graph crosses the x - axis at \(x=- 1\)? Wait, no, let's look at the graph. The graph has a V - shape? Wait, no, the left line is a straight line, then a V? Wait, the graph: the left line comes from the top left, crosses the y - axis, then crosses the x - axis at \(x = 1\)? Wait, no, let's re - examine. Let's count the grid squares. Let's assume each grid square is 1 unit. The left line: when \(x = 0\), \(y=1\) (y - intercept). Then the line goes down, crosses the x - axis at \(x = 1\)? Wait, no, the V - shaped part: the vertex is at \((2,-1)\)? Wait, maybe I made a mistake. Wait, the graph: the left line is a straight line with a negative slope, crosses the y - axis at \((0,1)\), then crosses the x - axis at \(x = 1\)? Then the right line has a positive slope, crossing the x - axis at \(x = 3\)? Wait, no, the original graph: let's see, the left line: from the top left, going down, crosses the y - axis at (0,1), then crosses the x - axis at (1,0), then the graph goes down to a vertex at (2, - 1), then up, crossing the x - axis at (3,0). Wait, maybe my initial assumption was wrong. Let's correctly identify:

x - intercepts: The points where \(y = 0\). From the graph, the graph crosses the x - axis at \(x = 1\) and \(x=3\)? Wait, no, looking at the grid, let's assume the origin is at (0,0). The left line: when \(x=-1\), \(y = 2\)? No, the y - axis is at \(x = 0\). Let's look at the graph again. The left - most part of the graph (the line with negative slope) crosses the x - axis at \(x=-1\)? No, the line crosses the y - axis at (0,1), then crosses the x - axis at (1,0), then the graph goes down to a minimum at (2, - 1), then up, crossing the x - axis at (3,0). So x - intercepts are \(x = 1\) and \(x = 3\).

y - intercept: The point where the graph crosses the y - axis (where \(x = 0\)). From the graph, when \(x = 0\), \(y = 1\), so the y - intercept is \(y = 1\) (the point \((0,1)\)).

Positive region: The function is positive when \(y>0\). So we look at the parts of the graph where the y - values are greater than 0. The left line: for \(x < 1\) (since it crosses the x - axis at \(x = 1\)) and the right line: for \(x>3\) (since it crosses the x - axis at \(x = 3\)), and between \(x = 1\) and \(x = 3\), the function is negative (below the x - axis). Wait, no, the vertex is at (2, - 1), so between \(x = 1\) and \(x = 3\), \(y<0\), and for \(x < 1\) and \(x>3\), \(y>0\).

Wait, let's correct:

x - intercepts: The graph crosses the x - axis at \(x = 1\) and \(x = 3\) (so the x - intercepts are \((1,0)\) and \((3,0)\), so \(x = 1\) and \(x = 3\)).

y - intercept: The graph crosses the y - axis at \((0,1)\), so \(y = 1\) (the y - intercept is 1).

Positive: The function is positive when \(x<1\) or \(x>3\) (because in these intervals, the graph is above the x - axis, \(y>0\)).

Negative: The function is negative when \(1 < x<3\) (because in this interval, the graph is below the x - axis, \(y < 0\)).

Wait, maybe the x - intercepts are \(x=-1\) and \(x = 3\)? No, let's re - check the graph. Let's assume the grid has a scale of 1 unit per square. The left line: from the top left, when \(x=-1\), \(y = 2\), then at \(x = 0\), \(y = 1\) (y - intercept), then at \(x = 1\), \(y = 0\) (x - intercept), then the graph goes down to \(x = 2\), \(y=-1\) (vertex), then up, at \(x = 3\), \(y = 0\) (x - intercept). So:

x - intercepts: \(x = 1\) and \(x = 3\) (the points \((1,0)\) and \((3,0)\)).

y - intercept: \(y = 1\) (the point \((0,1)\)).

Positive: When \(x<1\) or \(x>3\) (because for \(x<1\), the graph is above the x - axis, and for \(x>3\), the graph is above the x - axis).

Negative: When \(1 < x<3\) (because between \(x = 1\) and \(x = 3\), the graph is below the x - axis).

Step2: y - intercept

The y - intercept is the value of \(y\) when \(x = 0\). From the graph, when \(x = 0\), the graph is at \(y = 1\), so the y - intercept is \(y = 1\) (or the point \((0,1)\)).

Step3: Positive region

A function is positive when its graph is above the x - axis (\(y>0\)). From the graph, the graph is above the x - axis when \(x<1\) (to the left of the first x - intercept) and when \(x>3\) (to the right of the second x - intercept).

Step4: Negative region

A function is negative when its graph is below the x - axis (\(y < 0\)). From the graph, the graph is below the x - axis between the two x - intercepts, i.e., when \(1<x<3\).

Answer:

x - intercept(s): \(x = 1\) and \(x = 3\)

y - intercept: \(y = 1\)

positive: when \(x<1\) or \(x>3\)

negative: when \(1 < x<3\)

(Note: The answer may vary slightly depending on the exact scale of the graph, but based on the general shape, this is the correct analysis.)