Question

which statements correctly describe the graph of the function $f(x) = x^3 - 2x^2 - 19x + 20$? select three options.

• as the $x$-values increase, the $y$-values always increase.
• as $x$ approaches negative infinity, $y$ approaches negative infinity.
• the domain of the function is all real numbers.
• the range of the function is $y \geq 20$.
• the graph has a positive $y$-intercept.

Explanation:

Step1: Analyze the function type

The function \( f(x) = x^3 - 2x^2 - 19x + 20 \) is a cubic function (degree 3). For cubic functions, we analyze each statement:

Statement 1: "As the x - values increase, the y - values always increase."

The derivative of \( f(x) \) is \( f^\prime(x)=3x^2 - 4x - 19 \). The derivative is a quadratic function. The discriminant of \( f^\prime(x) \) is \( \Delta=(-4)^2-4\times3\times(-19)=16 + 228 = 244>0 \). So the derivative has two real roots, which means the function \( f(x) \) has intervals where it is increasing and decreasing. So this statement is false.

Statement 2: "As \( x \) approaches negative infinity, \( y \) approaches negative infinity."

For a cubic function \( y = ax^3+bx^2+cx + d \) with \( a = 1>0 \), the end - behavior is determined by the leading term \( ax^3 \). When \( x
ightarrow-\infty \), \( x^3
ightarrow-\infty \) (since \( a = 1>0 \)), so \( y
ightarrow-\infty \). This statement is true.

Statement 3: "The domain of the function is all real numbers."

Polynomial functions (including cubic functions) are defined for all real values of \( x \). So the domain of \( f(x) \) is \( (-\infty,\infty) \), i.e., all real numbers. This statement is true.

Statement 4: "The range of the function is \( y\geq20 \)."

Since the function is a cubic function (odd - degree), as \( x
ightarrow\infty \), \( y
ightarrow\infty \) and as \( x
ightarrow-\infty \), \( y
ightarrow-\infty \). So the range of the function is all real numbers, not \( y\geq20 \). This statement is false.

Statement 5: "The graph has a positive y - intercept."

The y - intercept is found by setting \( x = 0 \). Then \( f(0)=0^3-2\times0^2-19\times0 + 20=20>0 \). So the y - intercept is 20, which is positive. This statement is true.

Answer:

• As \( x \) approaches negative infinity, \( y \) approaches negative infinity.
• The domain of the function is all real numbers.
• The graph has a positive y - intercept.