Question

1. the graph shows the functions $f(x) = x^4 + 2x^3 - 13x^2 - 14x + 24$. find the following:

a. number of terms
b. leading coefficient
c. turning point(s)
d. relative minimum(s)
e. relative maximum(s)
f. degree
g. end behavior
h. $x-$intercept(s)

Explanation:

Part a: Number of terms

Step1: Identify terms in polynomial

A term is a single number, variable, or product of numbers and variables. In \( f(x) = x^4 + 2x^3 - 13x^2 - 14x + 24 \), the terms are \( x^4 \), \( 2x^3 \), \( -13x^2 \), \( -14x \), \( 24 \).

Step2: Count the terms

Counting these, we get 5 terms.

Part b: Leading Coefficient

Step1: Identify leading term

The leading term is the term with the highest degree. For \( f(x) = x^4 + 2x^3 - 13x^2 - 14x + 24 \), the leading term is \( x^4 \).

Step2: Find coefficient of leading term

The coefficient of \( x^4 \) is 1 (since \( x^4 = 1 \cdot x^4 \)).

Part c: Turning point(s)

Step1: Recall definition of turning points

Turning points are points where the graph changes from increasing to decreasing or vice - versa. For a polynomial of degree \( n \), the maximum number of turning points is \( n - 1 \). Here, degree is 4, so max turning points is 3.

Step2: Analyze the graph

From the graph, we can see the graph has 3 turning points. To find their coordinates, we look at the peaks and valleys. The left - most turning point (valley) is around \( x=-3 \), \( y = - 20\) (approx), the middle turning point (peak) is at \( x = 0\), \( y=30 \), and the right - most turning point (valley) is around \( x = 2\), \( y=-20\) (approx).

Part d: Relative Minimum(s)

Step1: Recall definition of relative minimum

A relative minimum is a point where the function changes from decreasing to increasing (a valley in the graph).

Step2: Identify relative minima from graph

From the graph, the relative minima occur at the two valley - like points. Their coordinates are approximately \( (-3, - 20) \) and \( (2, - 20) \).

Part e: Relative Maximum(s)

Step1: Recall definition of relative maximum

A relative maximum is a point where the function changes from increasing to decreasing (a peak in the graph).

Step2: Identify relative maximum from graph

From the graph, the relative maximum occurs at the peak - like point, which is at \( (0, 30) \).

Part f: Degree

Step1: Recall definition of degree of polynomial

The degree of a polynomial is the highest power of the variable in the polynomial.

Step2: Find degree of \( f(x) \)

For \( f(x)=x^4 + 2x^3 - 13x^2 - 14x + 24 \), the highest power of \( x \) is 4. So the degree is 4.

Part g: End Behavior

Step1: Recall end - behavior rules for polynomials

For a polynomial \( f(x)=a_nx^n+\cdots+a_0 \), if \( n \) is even:

• If \( a_n>0 \), as \( x

ightarrow\infty \), \( f(x)
ightarrow\infty \) and as \( x
ightarrow-\infty \), \( f(x)
ightarrow\infty \)

• If \( a_n<0 \), as \( x

ightarrow\infty \), \( f(x)
ightarrow-\infty \) and as \( x
ightarrow-\infty \), \( f(x)
ightarrow-\infty \)

Step2: Apply to given polynomial

Here, \( n = 4 \) (even) and leading coefficient \( a_n = 1>0 \). So as \( x
ightarrow\infty \), \( f(x)
ightarrow\infty \) and as \( x
ightarrow-\infty \), \( f(x)
ightarrow\infty \).

Part h: \( x \) - intercept(s)

Step1: Recall definition of \( x \) - intercept

\( x \) - intercepts are the points where \( y = 0 \) (where the graph crosses the \( x \) - axis).

Step2: Identify \( x \) - intercepts from graph and polynomial

From the graph, the graph crosses the \( x \) - axis at \( x=-4 \), \( x = - 2\), \( x = 1\), \( x = 3 \) (we can also factor the polynomial \( f(x)=(x + 4)(x + 2)(x - 1)(x - 3) \) to confirm). So the \( x \) - intercepts are \( (-4,0) \), \( (-2,0) \), \( (1,0) \), \( (3,0) \).

Answer:

s:
a. \(\boldsymbol{5}\)
b. \(\boldsymbol{1}\)
c. 3 turning points, approximately \((-3, - 20)\), \((0,30)\), \((2, - 20)\)
d. \((-3, - 20)\) and \((2, - 20)\) (approx)
e. \((0, 30)\)
f. \(\boldsymbol{4}\)
g. As \( x
ightarrow\pm\infty \), \( f(x)
ightarrow\infty \)
h. \((-4,0)\), \((-2,0)\), \((1,0)\), \((3,0)\)