Lesson 25 (10/28/26)

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Lesson 25 (10/28/26)

Graphing Functions - II

Warmup: Determine whether the following are odd or even

A collection of six function plots arranged in two rows. The top row shows a parabola symmetric about the y-axis labeled 'even', a parabola shifted horizontally to the right labeled 'Neither even nor odd', and a cubic-like curve symmetric about the origin labeled 'odd'. The bottom row shows a periodic wave function symmetric about the y-axis labeled 'even', and a periodic wave function symmetric about the origin labeled 'odd'. Below the diagrams, text defines 'Even' as f(-x) = f(x) with Y-axis symmetry, and 'odd' as f(-x) = -f(x) with Origin symmetry.
Visual Description: A collection of six function plots arranged in two rows. The top row shows a parabola symmetric about the y-axis labeled 'even', a parabola shifted horizontally to the right labeled 'Neither even nor odd', and a cubic-like curve symmetric about the origin labeled 'odd'. The bottom row shows a periodic wave function symmetric about the y-axis labeled 'even', and a periodic wave function symmetric about the origin labeled 'odd'. Below the diagrams, text defines 'Even' as f(-x) = f(x) with Y-axis symmetry, and 'odd' as f(-x) = -f(x) with Origin symmetry.

Even: \( f(-x) = f(x) \)

Y-axis symmetry

odd! \( f(-x) = -f(x) \)

Origin symmetry

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Sketching Guidelines, \( y = f(x) \)

Domain

x-intercepts and y-intercepts: For x-intercepts set \( y=0 \), for y-intercepts set \( x=0 \).

Vertical Asymptotes: \( \lim_{x \to a^+} f(x) = \pm \infty \) OR \( \lim_{x \to a^-} f(x) = \pm \infty \).

End Behaviour: \( \lim_{x \to \pm \infty} f(x) \). This helps identify Horizontal Asymptotes, Slant Asymptotes, or No Asymptote.

\( f'(x) \): \( f(x) \uparrow \) if \( f'(x) > 0 \), \( f(x) \downarrow \) if \( f'(x) < 0 \). Points where \( f'(x) = 0 \) or DNE are critical points.

\( f''(x) \): \( f''(x) > 0 \) is concave up, \( f''(x) < 0 \) is concave down. Points of inflection.

Symmetry: Even function - y-axis symmetry, odd function - origin symmetry.

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Example: Sketch the graph of \( f(x) = e^{x^2} \)

Domain: \( (-\infty, \infty) \)

Symmetry: \( f(-x) = e^{(-x)^2} = e^{x^2} = f(x) \). This is an even function and has y-axis symmetry.

Note: We can just graph on \( [0, \infty) \) and take the mirror image with respect to the y-axis.

X-intercept: Function value = 0 \(\implies\) \( e^{x^2} = 0 \). None, no x-intercept.

Y-intercept: \( y = f(0) = e^{0^2} = 1 \)

Vertical Asymptotes: \( f(x) \) has domain \( (-\infty, \infty) \) \(\implies\) No V.A.

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Horizontal Asymptotes / end behaviour:

\[ \lim_{x \to \infty} f(x) = \lim_{x \to \infty} e^{-x^2} = 0 \] \[ \lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} e^{-x^2} = 0 \] Both limits imply that \( y=0 \) is the Horizontal Asymptote.

\[ f'(x) = \frac{d}{dx} [e^{-x^2}] = e^{-x^2} \cdot [-2x] = -(2e^{-x^2}) \cdot x \] Note: The term \( (2e^{-x^2}) > 0 \).

A sign chart for the derivative f'(x). A horizontal line represents the x-axis with a point labeled 0 in the middle. To the left of 0, there is an arrow pointing up and the text 'f(x) is increasing on (-\infty, 0)'. To the right of 0, there is an arrow pointing down and the text 'f(x) is decreasing on (0, \infty)'. A bracket groups these intervals with the text 'local max at x=0'.
Visual Description: A sign chart for the derivative f'(x). A horizontal line represents the x-axis with a point labeled 0 in the middle. To the left of 0, there is an arrow pointing up and the text 'f(x) is increasing on (-\infty, 0)'. To the right of 0, there is an arrow pointing down and the text 'f(x) is decreasing on (0, \infty)'. A bracket groups these intervals with the text 'local max at x=0'.

The sign analysis shows:

  • \( f'(x) = -2(e^{-x^2})x > 0 \) for \( x < 0 \)
  • \( f'(x) = -2(e^{-x^2})x < 0 \) for \( x > 0 \)

As indicated by the diagram, \( f(x) \) is increasing on \( (-\infty, 0) \) and decreasing on \( (0, \infty) \), resulting in a local max at \( x=0 \).

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\(f''(x)\)!

\[ \frac{d}{dx} f'(x) = \frac{d}{dx} [-2e^{-x^2} \cdot x] \]

\[ = -2 \cdot \left( \frac{d}{dx} e^{-x^2} \right) \cdot x - 2e^{-x^2} \cdot \frac{d}{dx} x \]

\[ = (-2)(e^{-x^2} \cdot (-2x)) \cdot x - 2e^{-x^2} \cdot 1 \]

\[ = 2e^{-x^2} [2x^2 - 1] = 4e^{-x^2} [x^2 - 1/2] \]

Note: Under the expression \(e^{-x^2}\) there is an annotation indicating \(>0\).

A sign chart (number line diagram) for the expression x^2 - 1/2. The horizontal axis is marked with critical points x = -1/√2 and x = 1/√2. The intervals are defined as follows: For x < -1/√2, the expression x^2 - 1/2 > 0, leading to f'' > 0 and concave up (c. up). For -1/√2 < x < 1/√2, the expression x^2 - 1/2 < 0, leading to f'' < 0 and concave down (c. down). For x > 1/√2, the expression x^2 - 1/2 > 0, leading to f'' > 0 and concave up (c. up).
Visual Description: A sign chart (number line diagram) for the expression x^2 - 1/2. The horizontal axis is marked with critical points x = -1/√2 and x = 1/√2. The intervals are defined as follows: For x < -1/√2, the expression x^2 - 1/2 > 0, leading to f'' > 0 and concave up (c. up). For -1/√2 < x < 1/√2, the expression x^2 - 1/2 < 0, leading to f'' < 0 and concave down (c. down). For x > 1/√2, the expression x^2 - 1/2 > 0, leading to f'' > 0 and concave up (c. up).

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\( f(x) = e^{-x^2} \)

A hand-drawn graph of the Gaussian function f(x) = e^{-x^2} on a Cartesian coordinate system with grid lines. The curve is a bell-shaped symmetric curve centered on the y-axis. The peak of the curve occurs at the point (0, 1) on the y-axis, marked with the value '1'. The x-axis shows two marked points on either side of the origin, labeled as '-1/√2' and '1/√2'. The curve approaches the x-axis asymptotically as x increases and decreases away from the origin.
Visual Description: A hand-drawn graph of the Gaussian function f(x) = e^{-x^2} on a Cartesian coordinate system with grid lines. The curve is a bell-shaped symmetric curve centered on the y-axis. The peak of the curve occurs at the point (0, 1) on the y-axis, marked with the value '1'. The x-axis shows two marked points on either side of the origin, labeled as '-1/√2' and '1/√2'. The curve approaches the x-axis asymptotically as x increases and decreases away from the origin.

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Eg: Sketch the graph of \( y = \sin x - x \) on \( [-2\pi, 2\pi] \)

Symmetry: \( f(x) = \sin x - x \)

\[ f(-x) = \sin(-x) - (-x) = -\sin x + x = -[\sin x - x] = -f(x) \]

Note: Odd function. Graph has origin symmetry.

X-intercept: \( f(x) = 0 \implies \sin x - x = 0 \implies \sin x = x \)

Note: \( x = 0 \) is the only such number.

Y-intercept: \( f(0) = \sin 0 - 0 = 0 \)

End Behaviour: Do not have to consider.

Vertical Asymptotes: NONE

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Definition:

Note: when \( \cos x = 1 \)

A coordinate system plot showing the cosine function y = cos(x). The horizontal axis is marked with -2Ï€, 0, 2Ï€, and 4Ï€. A dashed vertical line segment indicates a peak at (0, 1).
Visual Description: A coordinate system plot showing the cosine function y = cos(x). The horizontal axis is marked with -2Ï€, 0, 2Ï€, and 4Ï€. A dashed vertical line segment indicates a peak at (0, 1).

Observation: \( f' = \cos x - 1 < 0 \)

Observation: \( f' = \cos x - 1 < 0 \)

A number line representing intervals on the x-axis, labeled with -2Ï€, 0, and 2Ï€. The plot shows the intervals where the derivative f' < 0, indicating decreasing function behavior.
Visual Description: A number line representing intervals on the x-axis, labeled with -2Ï€, 0, and 2Ï€. The plot shows the intervals where the derivative f' < 0, indicating decreasing function behavior.

decreasing on \( (-2\pi, 0) \) & \( (0, 2\pi) \)

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\[ f''(x) = \frac{d}{dx} f'(x) = \frac{d}{dx} [\cos x - 1] = -\sin x \]

A Cartesian coordinate system plotting the function y = sin(x). The x-axis is marked with intervals at -2π, -π, 0, π, and 2π. The curve starts at 0, oscillates downwards to a minimum between -2π and -π, crosses the origin (0,0), rises to a maximum between 0 and π, crosses the x-axis at π, goes to a minimum between π and 2π, and returns to 0 at 2π. The label 'sin x' is written above the curve.
Visual Description: A Cartesian coordinate system plotting the function y = sin(x). The x-axis is marked with intervals at -2π, -π, 0, π, and 2π. The curve starts at 0, oscillates downwards to a minimum between -2π and -π, crosses the origin (0,0), rises to a maximum between 0 and π, crosses the x-axis at π, goes to a minimum between π and 2π, and returns to 0 at 2π. The label 'sin x' is written above the curve.
A number line representing the interval from -2π to 2π, divided into segments based on the sign of -sin x and the resulting concavity of the function f''. The intervals are: from -2π to -π, -sin x < 0 and f'' < 0 (concave down); from -π to 0, -sin x > 0 and f'' > 0 (concave up); from 0 to π, -sin x < 0 and f'' < 0 (concave down); from π to 2π, -sin x > 0 and f'' > 0 (concave up). 'c. down' is used for concave down and 'c. up' is used for concave up.
Visual Description: A number line representing the interval from -2π to 2π, divided into segments based on the sign of -sin x and the resulting concavity of the function f''. The intervals are: from -2π to -π, -sin x < 0 and f'' < 0 (concave down); from -π to 0, -sin x > 0 and f'' > 0 (concave up); from 0 to π, -sin x < 0 and f'' < 0 (concave down); from π to 2π, -sin x > 0 and f'' > 0 (concave up). 'c. down' is used for concave down and 'c. up' is used for concave up.

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A Cartesian coordinate system plotted on a grid. The x-axis is marked with intervals at -2π, -π, π, and 2π. A continuous, monotonically decreasing blue curve passes through the origin (0,0). The curve is convex for x < 0 and concave for x > 0. Dotted vertical lines extend from the curve to the x-axis at the points x = -2π, x = -π, x = π, and x = 2π, indicating corresponding function values on the x-axis.
Visual Description: A Cartesian coordinate system plotted on a grid. The x-axis is marked with intervals at -2π, -π, π, and 2π. A continuous, monotonically decreasing blue curve passes through the origin (0,0). The curve is convex for x < 0 and concave for x > 0. Dotted vertical lines extend from the curve to the x-axis at the points x = -2π, x = -π, x = π, and x = 2π, indicating corresponding function values on the x-axis.

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Example: \( f'(x) = (x+4)(x-1)(x-5) \), sketch \( f(x) \)

Critical points: \( x=1 \), \( x=5 \), \( x=-4 \)

A sign chart diagram for the derivative f'(x). The number line shows the critical points at -4, 1, and 5. The intervals are marked with signs and arrows: For x < -4, f' < 0 and the function is decreasing (downward arrow). For -4 < x < 1, f' > 0 and the function is increasing (upward arrow). For 1 < x < 5, f' < 0 and the function is decreasing (downward arrow). For x > 5, f' > 0 and the function is increasing (upward arrow).
Visual Description: A sign chart diagram for the derivative f'(x). The number line shows the critical points at -4, 1, and 5. The intervals are marked with signs and arrows: For x < -4, f' < 0 and the function is decreasing (downward arrow). For -4 < x < 1, f' > 0 and the function is increasing (upward arrow). For 1 < x < 5, f' < 0 and the function is decreasing (downward arrow). For x > 5, f' > 0 and the function is increasing (upward arrow).

Increasing: \( (-4, 1) \), \( (5, \infty) \)

Decreasing: \( (-\infty, -4) \), \( (1, 5) \)

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\[ f'' = \frac{d}{dx} f'(x) = \frac{d}{dx} \left[ (x+4)(x-1)(x-5) \right] \]

\[ = \left[ \frac{d}{dx} (x+4) \right] (x-1)(x-5) + (x+4) \left[ \frac{d}{dx} (x-1)(x-5) \right] \]

Note: Under the first bracket, the derivative is \( 1 \). Under the second bracket, the expansion is \( \left( \frac{d}{dx}(x-1) \right)(x-5) + (x-1) \frac{d}{dx}(x-5) \), where each derivative is \( 1 \).

\[ = (x-1)(x-5) + (x+4)(x-5) + (x+4)(x-1) \]

\[ = x^2 - 6x + 5 + x^2 - x - 20 + x^2 + 3x - 4 \]

\[ = 3x^2 - 4x - 19 \]

Inflections

\[ 3x^2 - 4x - 19 = 0 \]

Note: This is a quadratic equation of the form \( ax^2 + bx + c = 0 \), which can be solved using the quadratic formula: \[ \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

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The equation \( 3x^2 - 4x - 19 = 0 \) is solved using the quadratic formula:

\[ \frac{4 \pm \sqrt{16 + 4(3)(-19)}}{6} \]

The roots are calculated as:

\[ x_1 = \frac{2 + \sqrt{61}}{3}, \quad x_2 = \frac{2 - \sqrt{61}}{3} \]

A Cartesian graph showing a downward-opening curve (likely a polynomial or sinusoidal-like wave). The horizontal axis shows marked values at -4, x_2, 1, x_1, and 5. The curve crosses the horizontal axis at x_2 and x_1. The curve reaches a local maximum between x_2 and 1 and a local minimum after x_1. A vertical axis intersects the horizontal axis near 0.
Visual Description: A Cartesian graph showing a downward-opening curve (likely a polynomial or sinusoidal-like wave). The horizontal axis shows marked values at -4, x_2, 1, x_1, and 5. The curve crosses the horizontal axis at x_2 and x_1. The curve reaches a local maximum between x_2 and 1 and a local minimum after x_1. A vertical axis intersects the horizontal axis near 0.