Question

ticket
name: ________________________ period: ____

1. describe a method for sketching the graph of a function given key features.

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:

Question 1: Describe a method for sketching the graph of a function given key features.

Step 1: Identify Key Features

List out key features: x - intercepts (where \(y = 0\)), y - intercept (where \(x = 0\)), relative maxima/minima (peaks and valleys), and end - behavior (what \(f(x)\) does as \(x
ightarrow\pm\infty\)).

Step 2: Plot Key Points

Mark the x - intercepts (e.g., \((a,0)\) for each x - intercept \(x = a\)), y - intercept \((0,b)\), and relative extrema (e.g., \((h,k)\) for a relative max/min at \(x = h\)) on the coordinate plane.

Step 3: Determine Intervals of Increase/Decrease

Using the relative maxima and minima, determine the intervals where the function is increasing (as \(x\) increases, \(f(x)\) increases) or decreasing (as \(x\) increases, \(f(x)\) decreases). For example, if there is a relative max at \(x = m\) and a relative min at \(x = n\) (\(m\lt n\)), the function increases from \(-\infty\) to \(m\), decreases from \(m\) to \(n\), and then increases from \(n\) to \(\infty\) (this is a general case and depends on the actual extrema).

Step 4: Connect Points Smoothly

Connect the plotted points with a smooth curve, following the intervals of increase and decrease. Make sure the curve passes through the intercepts and extrema, and follows the end - behavior. If the end - behavior is that \(f(x)\) increases as \(x
ightarrow\pm\infty\), the ends of the graph should go up as we move left (towards \(x
ightarrow-\infty\)) and right (towards \(x
ightarrow\infty\)).

Question 2: Sketch \(f(x)\) and \(g(x)\)

Sketching \(f(x)\)

Step 1: Plot Key Points

• X - intercepts: Plot the points \((- 5,0)\), \((0,0)\), \((2,0)\), \((4,0)\).
• Relative extrema: Plot the relative minimum at \((-3,-7)\) and \((4, - 2)\), and the relative maximum at \((1,1)\).
• Y - intercept: The y - intercept is \((0,0)\), which is already plotted as an x - intercept.

Step 2: Determine Intervals of Increase/Decrease

• From \(x

ightarrow-\infty\) to \(x=-3\): Since the next extremum is a relative minimum at \(x = - 3\) and then a relative maximum at \(x = 1\), the function should be increasing (because as \(x\) increases from \(-\infty\) to \(-3\), we are moving towards the minimum, but the end - behavior says \(f(x)\) increases as \(x\) increases or decreases. Wait, the problem says "As \(x\) increases or decreases, \(f(x)\) increases" which means as \(x
ightarrow\infty\), \(f(x)
ightarrow\infty\) and as \(x
ightarrow-\infty\), \(f(x)
ightarrow\infty\). So from \(x
ightarrow-\infty\) to \(x=-3\) (the first minimum), the function is increasing (since it goes from \(\infty\) to \(-7\)? Wait, no, maybe the description "As \(x\) increases or decreases, \(f(x)\) increases" means the end - behavior: \(\lim_{x
ightarrow\infty}f(x)=\infty\) and \(\lim_{x
ightarrow-\infty}f(x)=\infty\). So between \(x=-5\) and \(x = - 3\) (the first minimum), the function is decreasing from \((-5,0)\) to \((-3,-7)\)? Wait, maybe I misread. Let's re - examine: the problem says "As \(x\) increases or decreases, \(f(x)\) increases" – this is the end - behavior. So as \(x\) gets very large (positive) or very large (negative), \(f(x)\) increases. So for the intervals between the extrema:

• From \(x=-3\) (relative min) to \(x = 1\) (relative max): the function is increasing (from \(-7\) to \(1\)).
• From \(x = 1\) (relative max) to \(x = 4\) (relative min): the function is decreasing (from \(1\) to \(-2\)).
• From \(x = 4\) (relative min) to \(x

ightarrow\infty\): the function is increasing (since end - behavior is increasing as \(x
ightarrow\infty\)).

• From \(x

ightarrow-\infty\) to \(x=-3\) (relative min): the func…

Answer:

Step 1: Identify Key Features

List out key features: x - intercepts (where \(y = 0\)), y - intercept (where \(x = 0\)), relative maxima/minima (peaks and valleys), and end - behavior (what \(f(x)\) does as \(x
ightarrow\pm\infty\)).

Step 2: Plot Key Points

Mark the x - intercepts (e.g., \((a,0)\) for each x - intercept \(x = a\)), y - intercept \((0,b)\), and relative extrema (e.g., \((h,k)\) for a relative max/min at \(x = h\)) on the coordinate plane.

Step 3: Determine Intervals of Increase/Decrease

Using the relative maxima and minima, determine the intervals where the function is increasing (as \(x\) increases, \(f(x)\) increases) or decreasing (as \(x\) increases, \(f(x)\) decreases). For example, if there is a relative max at \(x = m\) and a relative min at \(x = n\) (\(m\lt n\)), the function increases from \(-\infty\) to \(m\), decreases from \(m\) to \(n\), and then increases from \(n\) to \(\infty\) (this is a general case and depends on the actual extrema).

Step 4: Connect Points Smoothly

Connect the plotted points with a smooth curve, following the intervals of increase and decrease. Make sure the curve passes through the intercepts and extrema, and follows the end - behavior. If the end - behavior is that \(f(x)\) increases as \(x
ightarrow\pm\infty\), the ends of the graph should go up as we move left (towards \(x
ightarrow-\infty\)) and right (towards \(x
ightarrow\infty\)).

Question 2: Sketch \(f(x)\) and \(g(x)\)

Sketching \(f(x)\)

Step 1: Plot Key Points

• X - intercepts: Plot the points \((- 5,0)\), \((0,0)\), \((2,0)\), \((4,0)\).
• Relative extrema: Plot the relative minimum at \((-3,-7)\) and \((4, - 2)\), and the relative maximum at \((1,1)\).
• Y - intercept: The y - intercept is \((0,0)\), which is already plotted as an x - intercept.

Step 2: Determine Intervals of Increase/Decrease

• From \(x

ightarrow-\infty\) to \(x=-3\): Since the next extremum is a relative minimum at \(x = - 3\) and then a relative maximum at \(x = 1\), the function should be increasing (because as \(x\) increases from \(-\infty\) to \(-3\), we are moving towards the minimum, but the end - behavior says \(f(x)\) increases as \(x\) increases or decreases. Wait, the problem says "As \(x\) increases or decreases, \(f(x)\) increases" which means as \(x
ightarrow\infty\), \(f(x)
ightarrow\infty\) and as \(x
ightarrow-\infty\), \(f(x)
ightarrow\infty\). So from \(x
ightarrow-\infty\) to \(x=-3\) (the first minimum), the function is increasing (since it goes from \(\infty\) to \(-7\)? Wait, no, maybe the description "As \(x\) increases or decreases, \(f(x)\) increases" means the end - behavior: \(\lim_{x
ightarrow\infty}f(x)=\infty\) and \(\lim_{x
ightarrow-\infty}f(x)=\infty\). So between \(x=-5\) and \(x = - 3\) (the first minimum), the function is decreasing from \((-5,0)\) to \((-3,-7)\)? Wait, maybe I misread. Let's re - examine: the problem says "As \(x\) increases or decreases, \(f(x)\) increases" – this is the end - behavior. So as \(x\) gets very large (positive) or very large (negative), \(f(x)\) increases. So for the intervals between the extrema:

• From \(x=-3\) (relative min) to \(x = 1\) (relative max): the function is increasing (from \(-7\) to \(1\)).
• From \(x = 1\) (relative max) to \(x = 4\) (relative min): the function is decreasing (from \(1\) to \(-2\)).
• From \(x = 4\) (relative min) to \(x

ightarrow\infty\): the function is increasing (since end - behavior is increasing as \(x
ightarrow\infty\)).

• From \(x

ightarrow-\infty\) to \(x=-3\) (relative min): the function is decreasing (since it goes from \(\infty\) to \(-7\))? But the end - behavior says as \(x
ightarrow-\infty\), \(f(x)\) increases. Wait, there is a contradiction unless the function has a different structure. Maybe the function is a polynomial of even degree (since end - behavior is the same on both ends, increasing) with multiple roots. Let's assume the function is a polynomial. The x - intercepts are at \(x=-5,0,2,4\), so the function can be written in factored form as \(f(x)=ax(x + 5)(x - 2)(x - 4)\) (since it passes through \((0,0)\)). Now, we can use the extrema to find \(a\), but for sketching, we can use the key points.

Step 3: Connect the Points

• Start from the left ( \(x

ightarrow-\infty\), \(f(x)
ightarrow\infty\) ), come down to the x - intercept \((-5,0)\), then down to the relative minimum at \((-3,-7)\), then up to the relative maximum at \((1,1)\), then down to the relative minimum at \((4,-2)\), then up through the x - intercept \((4,0)\) (wait, no, the x - intercept is at \((4,0)\) and the relative minimum is at \((4,-2)\)? That means the function touches or crosses the x - axis at \(x = 4\) and has a minimum there. So the graph touches the x - axis at \(x = 4\) (since the minimum is at \((4,-2)\) and the x - intercept is at \((4,0)\)? Wait, no, the x - intercept is where \(y = 0\), so \((4,0)\) is an x - intercept, and the relative minimum is at \((4,-2)\) – that means the function crosses the x - axis at \(x = 4\) and then goes down to \((4,-2)\)? No, the x - intercept is a point where \(y = 0\), so \((4,0)\) is on the x - axis, and the relative minimum is at \((4,-2)\) – that would mean the function has a minimum at \(x = 4\) below the x - axis, but also an x - intercept at \(x = 4\), which is a contradiction unless there is a double root at \(x = 4\). Maybe the x - intercepts are at \(x=-5,0,2\) and \(x = 4\) is a double root? Let's adjust. If \(x = 4\) is a double root, then the function is \(f(x)=ax(x + 5)(x - 2)(x - 4)^2\). Then at \(x = 4\), the function touches the x - axis (since it's a double root) and has a minimum there.

• So, starting from the left (\(x

ightarrow-\infty\), \(f(x)
ightarrow\infty\) because the leading coefficient \(a\) is positive, since the end - behavior is increasing as \(x
ightarrow\pm\infty\) and the degree is \(5\)? Wait, no, if we have \(x(x + 5)(x - 2)(x - 4)^2\), the degree is \(1 + 1+1 + 2=5\), which is odd, but the end - behavior would be \(\lim_{x
ightarrow\infty}f(x)=\infty\) and \(\lim_{x
ightarrow-\infty}f(x)=-\infty\) if \(a>0\), which contradicts the given end - behavior. So maybe the function is of even degree, so we need to have an even number of linear factors. Let's say \(f(x)=ax(x + 5)(x - 2)(x - 4)\) is degree 4 (even). Then \(\lim_{x
ightarrow\infty}f(x)=\infty\) and \(\lim_{x
ightarrow-\infty}f(x)=\infty\) if \(a>0\). Now, the x - intercepts are at \(x=-5,0,2,4\) (all simple roots). Then the relative extrema: we have a relative min at \((-3,-7)\), relative max at \((1,1)\), and relative min at \((4,-2)\). So the graph:

• From \(x

ightarrow-\infty\) (top left) comes down to \((-5,0)\), then down to \((-3,-7)\) (relative min), then up to \((1,1)\) (relative max), then down to \((4,-2)\) (relative min), then up to \((2,0)\) and then up to \(x
ightarrow\infty\) (top right). Wait, but we also have an x - intercept at \((0,0)\). So when \(x = 0\), \(y = 0\), which is on the graph. So the graph passes through \((0,0)\), \((-5,0)\), \((2,0)\), \((4,0)\), with minima at \((-3,-7)\) and \((4,-2)\), and a maximum at \((1,1)\).

Sketching \(g(x)\)

Step 1: Identify Common Features with \(f(x)\)

• X - intercepts: Same as \(f(x)\): \((-5,0)\), \((0,0)\), \((2,0)\), \((4,0)\).
• End - behavior: Same as \(f(x)\): as \(x

ightarrow\pm\infty\), \(g(x)\) increases.

Step 2: Adjust Extrema

• Greater relative maximum: The relative maximum of \(f(x)\) is at \((1,1)\). For \(g(x)\), we need a relative maximum with a higher \(y\) - value, say \((1,3)\) (any value greater than 1).
• Lesser relative minima: The relative minima of \(f(x)\) are at \((-3,-7)\) and \((4,-2)\). For \(g(x)\), we need relative minima with higher \(y\) - values (lesser in magnitude of the minimum). For example, the relative minimum at \((-3,-7)\) can be changed to \((-3,-3)\) (higher than - 7) and the relative minimum at \((4,-2)\) can be changed to \((4,0)\) (higher than - 2, and also on the x - axis, so it's a touch point).

Step 3: Plot New Key Points and Connect

• Plot the same x - intercepts: \((-5,0)\), \((0,0)\), \((2,0)\), \((4,0)\) (with \((4,0)\) now as a relative minimum).
• Plot the new relative maximum at \((1,3)\) and new relative minima at \((-3,-3)\) and \((4,0)\).
• Connect the points following the end - behavior (increasing as \(x

ightarrow\pm\infty\)) and the intervals of increase/decrease. From \(x
ightarrow-\infty\) to \(x=-3\), the function decreases from \(\infty\) to \((-3,-3)\), then increases to \((1,3)\), then decreases to \((4,0)\), then increases to \(x
ightarrow\infty\).

Final Answers (Descriptive for Sketches)

1. The method for sketching a function from key features involves identifying key points (intercepts, extrema), determining intervals of increase/decrease, and connecting points smoothly.
2. For \(f(x)\), plot the x - intercepts \((-5,0)\), \((0,0)\), \((2,0)\), \((4,0)\), extrema \((-3,-7)\), \((1,1)\), \((4,-2)\) and connect with a smooth curve following the end - behavior. For \(g(x)\), use the same x - intercepts and end - behavior, but plot a higher relative maximum (e.g., \((1,3)\)) and higher relative minima (e.g., \((-3,-3)\), \((4,0)\)) and connect the points.