Question

1. f-if.7

identify the vertical and horizontal intercepts of this graph. and describe the end behavior of this graph:

Explanation:

Step1: Identify Intercepts

• Vertical (y - intercept): The point where the graph crosses the y - axis. From the graph, when \(x = 0\), \(y=6\), so the y - intercept is \((0,6)\).
• Horizontal (x - intercepts): The points where the graph crosses the x - axis. From the graph and the given points, the x - intercepts are \((- 3,0)\), \((1,0)\), and \((2,0)\).

Step2: Analyze End Behavior

• For the end behavior of a polynomial graph, we look at the leading term. The graph rises to the right (as \(x

ightarrow+\infty\), \(y
ightarrow+\infty\)) and falls to the left? Wait, no, looking at the left - most part, as \(x
ightarrow-\infty\), the graph goes down? Wait, no, the left - most part: when \(x
ightarrow-\infty\), let's see the shape. Wait, the graph has a root at \(x = - 3\), and then a peak. But the right - hand end: as \(x
ightarrow+\infty\), the graph goes up (since the right - most part is increasing). For the left - hand end, as \(x
ightarrow-\infty\), let's check the direction. The graph at \(x=-3\) is a root, and to the left of \(x = - 3\), the graph is going down (since when \(x\) is less than \(-3\), the function value is decreasing, as we move left from \(x=-3\), the graph goes down). Wait, actually, the end behavior: as \(x
ightarrow+\infty\), \(y
ightarrow+\infty\) (since the right end is rising), and as \(x
ightarrow-\infty\), \(y
ightarrow-\infty\) (since the left end is falling). Wait, but let's confirm with the degree. The graph has three x - intercepts, but maybe it's a cubic? No, the graph has a "w" shape? Wait, no, the graph has a peak on the left, then a valley, then another valley, then rises. Wait, the key is: as \(x\) approaches positive infinity, the graph goes up (since the right - most part is increasing), and as \(x\) approaches negative infinity, the graph goes down (since the left - most part is decreasing).

Answer:

• Vertical (y - intercept): \((0,6)\)
• Horizontal (x - intercepts): \((-3,0)\), \((1,0)\), \((2,0)\)
• End Behavior: As \(x

ightarrow+\infty\), \(y
ightarrow+\infty\); as \(x
ightarrow-\infty\), \(y
ightarrow-\infty\)