Question

1. sketch ( f(x) ) given that the function has

x-intercepts at ( -5, 0, 2, ) and ( 4, )
relative minima at ( (-3, -7) ), and ( (4, -2) ) and a relative maximum at ( (1, 1) ).
( f(x) ) intersects the y-axis at ( (0, 0) ).
as ( x ) increases or decreases, ( f(x) ) increases.

sketch the graph of a function ( g(x) ) that has a greater relative maximum, lesser relative minima, and the
same x-intercepts and end behavior as ( f(x) ).

Explanation:

Sketching \( f(x) \)

Step 1: Plot Key Points

• x-intercepts: Mark points \((-5, 0)\), \((0, 0)\), \((2, 0)\), \((4, 0)\) on the \( x \)-axis.
• Relative Minima: Plot \((-3, -7)\) and \((4, -2)\).
• Relative Maximum: Plot \((1, 1)\).
• y-intercept: Already included in \( x \)-intercepts (\((0, 0)\)).

Step 2: Analyze End Behavior

As \( x \to \pm\infty \), \( f(x) \to +\infty \) (since \( f(x) \) increases for large \( |x| \)). This implies the leading term of \( f(x) \) has an even degree and positive leading coefficient.

Step 3: Connect Points with Smooth Curves

• Between \( x = -5 \) and \( x = -3 \): The function decreases from \( (-5, 0) \) to the relative minimum at \( (-3, -7) \) (since it reaches a minimum there).
• Between \( x = -3 \) and \( x = 1 \): The function increases from \( (-3, -7) \) to the relative maximum at \( (1, 1) \).
• Between \( x = 1 \) and \( x = 4 \): The function decreases from \( (1, 1) \) to the relative minimum at \( (4, -2) \) (note: \( (4, 0) \) is an \( x \)-intercept, so the curve passes through \( (4, 0) \) before reaching \( (4, -2) \)? Wait, no—\( x = 4 \) is an \( x \)-intercept, so the point \( (4, 0) \) is on the graph, and the relative minimum is at \( (4, -2) \)? Wait, that might be a typo, but assume the minimum is near \( x = 4 \).
• For \( x > 4 \): The function increases (end behavior: \( f(x) \to +\infty \)).
• For \( x < -5 \): The function increases (end behavior: \( f(x) \to +\infty \)).

Sketching \( g(x) \)

Step 1: Maintain Key Features

• x-intercepts: Same as \( f(x) \): \((-5, 0)\), \((0, 0)\), \((2, 0)\), \((4, 0)\).
• End Behavior: Same as \( f(x) \): \( g(x) \to +\infty \) as \( x \to \pm\infty \) (even degree, positive leading coefficient).

Step 2: Adjust Extrema

• Greater Relative Maximum: Let the new relative maximum be at \( (1, M) \) where \( M > 1 \) (e.g., \( M = 3 \)).
• Lesser Relative Minima: Let the new relative minima be at \( (-3, m_1) \) and \( (4, m_2) \) where \( m_1 > -7 \) and \( m_2 > -2 \) (e.g., \( m_1 = -5 \), \( m_2 = 0 \) (but \( (4, 0) \) is an intercept, so \( m_2 \) could be \( -1 \))).

Step 3: Connect Points with Smooth Curves

• Follow the same general shape as \( f(x) \), but adjust the heights of extrema:
• From \( x < -5 \): Increase to \( (-5, 0) \), then decrease to the new minimum at \( (-3, m_1) \), then increase to the new maximum at \( (1, M) \), then decrease to the new minimum at \( (4, m_2) \), then increase to \( +\infty \) as \( x \to +\infty \).

Final Sketch Notes

• For \( f(x) \), the graph will have "peaks" and "valleys" at the given extrema, passing through all \( x \)-intercepts.
• For \( g(x) \), the "peak" at \( x = 1 \) is taller, and the "valleys" at \( x = -3 \) and \( x = 4 \) are shallower, while still passing through the same \( x \)-intercepts and having the same end behavior.

(Note: Since sketching is visual, the key is to plot the points and connect them with smooth curves following the increasing/decreasing behavior and end behavior. For \( g(x) \), adjust the extrema heights while preserving intercepts and end behavior.)

Answer:

Sketching \( f(x) \)

Step 1: Plot Key Points

• x-intercepts: Mark points \((-5, 0)\), \((0, 0)\), \((2, 0)\), \((4, 0)\) on the \( x \)-axis.
• Relative Minima: Plot \((-3, -7)\) and \((4, -2)\).
• Relative Maximum: Plot \((1, 1)\).
• y-intercept: Already included in \( x \)-intercepts (\((0, 0)\)).

Step 2: Analyze End Behavior

As \( x \to \pm\infty \), \( f(x) \to +\infty \) (since \( f(x) \) increases for large \( |x| \)). This implies the leading term of \( f(x) \) has an even degree and positive leading coefficient.

Step 3: Connect Points with Smooth Curves

• Between \( x = -5 \) and \( x = -3 \): The function decreases from \( (-5, 0) \) to the relative minimum at \( (-3, -7) \) (since it reaches a minimum there).
• Between \( x = -3 \) and \( x = 1 \): The function increases from \( (-3, -7) \) to the relative maximum at \( (1, 1) \).
• Between \( x = 1 \) and \( x = 4 \): The function decreases from \( (1, 1) \) to the relative minimum at \( (4, -2) \) (note: \( (4, 0) \) is an \( x \)-intercept, so the curve passes through \( (4, 0) \) before reaching \( (4, -2) \)? Wait, no—\( x = 4 \) is an \( x \)-intercept, so the point \( (4, 0) \) is on the graph, and the relative minimum is at \( (4, -2) \)? Wait, that might be a typo, but assume the minimum is near \( x = 4 \).
• For \( x > 4 \): The function increases (end behavior: \( f(x) \to +\infty \)).
• For \( x < -5 \): The function increases (end behavior: \( f(x) \to +\infty \)).

Sketching \( g(x) \)

Step 1: Maintain Key Features

• x-intercepts: Same as \( f(x) \): \((-5, 0)\), \((0, 0)\), \((2, 0)\), \((4, 0)\).
• End Behavior: Same as \( f(x) \): \( g(x) \to +\infty \) as \( x \to \pm\infty \) (even degree, positive leading coefficient).

Step 2: Adjust Extrema

• Greater Relative Maximum: Let the new relative maximum be at \( (1, M) \) where \( M > 1 \) (e.g., \( M = 3 \)).
• Lesser Relative Minima: Let the new relative minima be at \( (-3, m_1) \) and \( (4, m_2) \) where \( m_1 > -7 \) and \( m_2 > -2 \) (e.g., \( m_1 = -5 \), \( m_2 = 0 \) (but \( (4, 0) \) is an intercept, so \( m_2 \) could be \( -1 \))).

Step 3: Connect Points with Smooth Curves

• Follow the same general shape as \( f(x) \), but adjust the heights of extrema:
• From \( x < -5 \): Increase to \( (-5, 0) \), then decrease to the new minimum at \( (-3, m_1) \), then increase to the new maximum at \( (1, M) \), then decrease to the new minimum at \( (4, m_2) \), then increase to \( +\infty \) as \( x \to +\infty \).

Final Sketch Notes

• For \( f(x) \), the graph will have "peaks" and "valleys" at the given extrema, passing through all \( x \)-intercepts.
• For \( g(x) \), the "peak" at \( x = 1 \) is taller, and the "valleys" at \( x = -3 \) and \( x = 4 \) are shallower, while still passing through the same \( x \)-intercepts and having the same end behavior.

(Note: Since sketching is visual, the key is to plot the points and connect them with smooth curves following the increasing/decreasing behavior and end behavior. For \( g(x) \), adjust the extrema heights while preserving intercepts and end behavior.)