Question

graphs of functions have many features that we want to become familiar with. the names of these features are often very descriptive. it is important to note that when we ask about values of a function then we are referring to the output or y - values.
exercise #1: the function ( y = f(x) ) is shown graphed below. answer the following questions.
(a) find the minimum and maximum values of the function. state the values of x where they occur as well.
minimum value = ______ at ( x = ) ______
maximum value = ______ at ( x = ) ______
(b) what is the y - intercept of the function?
what is the x - value at any functions y - intercept?
(c) circle and state the x - intercepts of the function. these are also known as the zeros of the function because they are where ( f(x)=0 ).
(d) would you characterize the function as increasing or decreasing over the interval ( -5lt xlt2 )? explain your choice.
(e) give one interval over which the function is increasing. use any appropriate notation.
(f) state the coordinates of all turning points on this function and label them as either a maximum or a minimum.
(g) give the interval over which the function is negative, i.e. ( f(x)lt0 ).

Explanation:

Step1: Analyze the graph for minima and maxima

The minimum value is the lowest \(y -\)value on the graph. By observing the graph, the lowest \(y -\)value (minimum) is \(- 7\) and it occurs at \(x = 2\). The maximum value is the highest \(y -\)value on the graph. The highest \(y -\)value (maximum) is \(8\) and it occurs at \(x=-6\).

Step2: Determine the \(y -\)intercept

The \(y -\)intercept of a function \(y = f(x)\) is the value of \(y\) when \(x = 0\). Looking at the graph, when \(x = 0\), \(y=-3\). For any function's \(y -\)intercept, the \(x -\)value is \(x = 0\) (by the definition of \(y -\)intercept: the point \((0,y)\) where the graph crosses the \(y -\)axis).

Step3: Find the \(x -\)intercepts

The \(x -\)intercepts are the values of \(x\) for which \(y=f(x)=0\). By looking at the graph, the graph crosses the \(x -\)axis at \(x=-2\) and \(x = 7\).

Step4: Check if the function is increasing or decreasing in \(-5\lt x\lt2\)

A function \(y = f(x)\) is decreasing if as \(x\) increases, \(y\) decreases. In the interval \(-5\lt x\lt2\), as \(x\) increases (moves from left to right on the \(x -\)axis), the \(y -\)values of the function decrease. So the function is decreasing in the interval \(-5\lt x\lt2\).

Step5: Find an increasing interval

A function \(y = f(x)\) is increasing if as \(x\) increases, \(y\) increases. Looking at the graph, the function is increasing in the interval \(2\lt x\lt10\) (for example).

Step6: Identify turning points

Turning points are points where the function changes from increasing to decreasing or vice - versa. The turning points are \((-6,8)\) (a maximum since the function changes from increasing to decreasing at this point) and \((2,-7)\) (a minimum since the function changes from decreasing to increasing at this point).

Step7: Find the interval where \(f(x)\lt0\)

We need to find the values of \(x\) for which \(y = f(x)\lt0\). By looking at the graph, the function \(y=f(x)\) is below the \(x -\)axis (where \(y\lt0\)) in the interval \(-2\lt x\lt7\).

Answer:

(a) minimum value \(=-7\) at \(x = 2\); maximum value \(=8\) at \(x=-6\)
(b) \(y -\)intercept \(=-3\); \(x -\)value at \(y -\)intercept \(=0\)
(c) \(x -\)intercepts: \(x=-2\) and \(x = 7\)
(d) The function is decreasing over the interval \(-5\lt x\lt2\) because as \(x\) increases in this interval, \(y\) decreases.
(e) One increasing interval: \((2,10)\) (using interval notation)
(f) Turning points: \((-6,8)\) (maximum) and \((2,-7)\) (minimum)
(g) The function is negative (\(f(x)\lt0\)) in the interval \((-2,7)\)