Question

identify each characteristic for the graph of the function shown.
select the number of y - intercepts:
enter the coordinates of the y - intercept:
select the number of x - intercepts:
enter the coordinates of the x - intercepts:
decide whether the function has extrema:
select a statement to describe the domain:
select a statement to describe the range:
enter an equation or inequality for the range:

Explanation:

Step1: Number of y - intercepts

The y - intercept is the point where the graph crosses the y - axis. For a polynomial function, the graph crosses the y - axis at one point (when \(x = 0\)). So the number of y - intercepts is 1.

Step2: Coordinates of y - intercept

To find the y - intercept, we set \(x=0\) and find the corresponding \(y\) - value. From the graph, when \(x = 0\), we look at the point where the graph intersects the y - axis. The graph passes through \((0,30)\) (by looking at the grid, the y - axis is at \(x = 0\) and the graph intersects the y - axis at \(y=30\)).

Step3: Number of x - intercepts

The x - intercepts are the points where the graph crosses the x - axis. From the graph, we can see that the graph crosses the x - axis at two points. So the number of x - intercepts is 2.

Step4: Coordinates of x - intercepts

Looking at the graph, the graph crosses the x - axis at \(x=- 4\) and \(x = 8\) (by looking at the grid, the x - axis is \(y = 0\)). So the x - intercepts are \((-4,0)\) and \((8,0)\).

Step5: Extrema of the function

The function is a parabola opening downwards (since it has a maximum point). So it has a relative maximum (and also an absolute maximum). The vertex of the parabola (the highest point) is the relative maximum. From the graph, the vertex is at \((2,35)\) (by looking at the grid, the x - coordinate of the vertex is 2 and the y - coordinate is 35). So the relative maximum (and absolute maximum) is at \((2,35)\).

Step6: Domain of the function

For a polynomial function (a parabola is a quadratic polynomial), the domain is all real numbers because there are no restrictions on the values of \(x\) that we can plug into the function.

Step7: Range of the function

Since the parabola opens downwards, the maximum value of \(y\) is the y - coordinate of the vertex (35) and \(y\) can take any value less than or equal to 35. So the range is all real numbers \(y\) such that \(y\leqslant35\).

Answer:

• Number of y - intercepts: 1
• Coordinates of y - intercept: \((0,30)\)
• Number of x - intercepts: 2
• Coordinates of x - intercepts: \((-4,0)\) and \((8,0)\)
• Extrema: Relative maximum (and absolute maximum) at \((2,35)\)
• Domain: All real numbers
• Range: \(y\leqslant35\) (or \(f(x)\leqslant35\))