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.
key features of functions
n - gen math algebra i
date:
exercise #1: the function ( y = f(x) ) is shown graphed below. answer the following questions.
(a) state 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 ( -5 < x < 2 )? 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)<0 ).

Explanation:

Step1: Identify minimum and maximum values

Minimum value is the lowest \(y\) - value. From the graph, the minimum \(y=-7\) at \(x = 2\). Maximum value is the highest \(y\) - value. From the graph, the maximum \(y = 9\) at \(x=- 4\).

Step2: Find \(y\) - intercept

The \(y\) - intercept is the value of \(y\) when \(x = 0\). From the graph, when \(x = 0\), \(y=-3\). The \(x\) - value at any function's \(y\) - intercept is \(x = 0\).

Step3: Determine \(x\) - intercepts

The \(x\) - intercepts are the values of \(x\) for which \(y=f(x)=0\). From the graph, \(x=-12\) and \(x = 7\).

Step4: Analyze function behavior in the interval \(-5\lt x\lt2\)

As \(x\) increases from \(-5\) to \(2\), the \(y\) - values of the function decrease. So the function is decreasing over the interval \(-5\lt x\lt2\).

Step5: Find an increasing interval

An increasing interval is where as \(x\) increases, \(y\) increases. For example, the interval \(2\lt x\lt10\) (there are other possible correct intervals like \(-10\lt x\lt - 4\)).

Step6: Identify turning points

Turning points are points where the function changes from increasing to decreasing or vice - versa. The turning points are \((-4,9)\) (maximum) and \((2,-7)\) (minimum).

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

The function \(y = f(x)\lt0\) when the graph is below the \(x\) - axis. From the graph, the interval is \(-12\lt x\lt7\).

Answer:

(a) minimum value \(=-7\) at \(x = 2\); maximum value \(=9\) at \(x=-4\)
(b) \(y\) - intercept \(=-3\); \(x\) - value at \(y\) - intercept \(x = 0\)
(c) \(x\) - intercepts \(x=-12\) and \(x = 7\)
(d) The function is decreasing over \(-5\lt x\lt2\) because as \(x\) increases, \(y\) decreases.
(e) One increasing interval is \(2\lt x\lt10\) (or \(-10\lt x\lt - 4\))
(f) Turning points: \((-4,9)\) (maximum) and \((2,-7)\) (minimum)
(g) The interval where \(f(x)\lt0\) is \(-12\lt x\lt7\)