Problem 52 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 47-52: Sketch the graph of \(f\). $$ f(x)= \begin{cases}x-3 & \text { if } x \leq-2 \\ -x^{2} & \text { if }-2

Step-by-Step Solution

Answer

Sketch separate segments: a line for \(x \leq -2\), parabola for \(-2

Step 1: Identify the piecewise function intervals

The function \(f(x)\) is defined differently over three intervals: \(x \leq -2\), \(-2 < x < 1\), and \(x \geq 1\). Each segment corresponds to a different formula. We'll need to treat each segment separately when sketching the graph.

Step 2: Analyze and sketch \(f(x) = x-3\) for \(x \leq -2\)

For \(x \leq -2\), the function is \(f(x) = x-3\), a straight line with a slope of 1 and y-intercept at \(x = 0\), \(f(x) = -3\). Since this equation is valid for \(x \leq -2\), plot this line starting at the point \((-2, -5)\) going leftwards. Include a closed circle at \((-2, -5)\) to indicate the endpoint is included.

Step 3: Analyze and sketch \(f(x) = -x^2\) for \(-2 < x < 1\)

For \(-2 < x < 1\), the function is \(f(x) = -x^2\), a downward-opening parabola. Since it spans from \(x > -2\) to \(x < 1\), calculate important values like the vertex and points at either interval end: at \(x = 0\), \(f(x) = 0\) (vertex); at \(x = -1\), \(f(x) = -1\); at \(x = 1\), \(f(x) = -1\). Plot these and draw a smooth curve with open endpoints at \((-2, -4)\) and \((1, -1)\). Include an open circle at both ends.

Step 4: Analyze and sketch \(f(x) = -x + 4\) for \(x \geq 1\)

For \(x \geq 1\), the function is \(f(x) = -x + 4\), a straight line with a slope of \(-1\) and y-intercept at \(x = 0\), \(f(x) = 4\). This segment begins at \((1, 3)\), and should be plotted with a closed circle since \(x = 1\) is included. Extend the line to the right. Include this segment in the graph.

Key Concepts

Graph Sketching

Graph sketching is like drawing a picture of a function based on its mathematical equations. It's an essential skill in understanding how different types of functions behave visually over certain ranges of inputs.
In this exercise, you are asked to sketch a piecewise function, which is a kind of function that has different rules or formulas for different parts of its domain.

• Start by identifying the different pieces or intervals of the function.
• Determine the formula that applies to each interval and plot them separately.
• Join these plots carefully while respecting where each rule applies, paying special attention to endpoints.

When graphing each segment, remember to use circles to indicate whether points are included or not. A closed circle implies that the endpoint is part of the graph, while an open circle implies it is not. Get creative with your sketch, but ensure your graph accurately represents the intervals and formulas given.

Function Intervals

Understanding function intervals is key when dealing with piecewise functions. Each interval tells you where a particular equation applies. In this context, intervals can be found in inequalities, like "for \( x \leq -2 \)" or "\( -2 < x < 1 \)."
To handle these intervals, follow these easy steps:

• Read each segment's inequality correctly to identify whether endpoints are included.
• Note the transitions between pieces so you can join them cleanly on your graph.

Each interval's behavior may differ. For instance, in the current piecewise function:

• For the interval \( x \leq -2 \), the graph is linear, and the endpoint \( (-2) \) is included, indicated by a filled circle.
• In the interval \( -2 < x < 1 \), the graph forms a parabola, but does not include endpoints \( x = -2 \) or \( x = 1 \), shown by open circles.
• Finally, for \( x \geq 1 \), begin at \( x = 1 \) and extend a linear graph to the right, with a closed circle at \( x = 1 \).

These intervals thus guide you on how to draw and connect the function's segments accurately.

Parabolas

Parabolas are important curves in mathematics, representing quadratic functions. The function in this exercise, \( f(x) = -x^2 \), is an example of a downward-opening parabola. This shape is determined by the negative sign in front of \( x^2 \).
When sketching parabolas, it helps to know some key points:

• The vertex is the peak (or turning point); for \( f(x) = -x^2 \), the vertex is at the origin, \( (0, 0) \).
• Symmetry means the parabola is mirror-like about its vertical line, here coinciding with the y-axis.
• Check values at the endpoints of the interval to understand its engagement without strictly including them—for example, \( -1 \) and \( 1 \) in this function return \( -1 \) each.

Understanding these points helps not just in drawing the curve, but also in predicting the behavior across intervals. Keeping each segment's rules in mind prevents mixing up parabolic and linear behaviors in the sketch.

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Key Concepts

Graph Sketching

Graph sketching is like drawing a picture of a function based on its mathematical equations. It's an essential skill in understanding how different types of functions behave visually over certain ranges of inputs.
In this exercise, you are asked to sketch a piecewise function, which is a kind of function that has different rules or formulas for different parts of its domain.

• Start by identifying the different pieces or intervals of the function.
• Determine the formula that applies to each interval and plot them separately.
• Join these plots carefully while respecting where each rule applies, paying special attention to endpoints.

When graphing each segment, remember to use circles to indicate whether points are included or not. A closed circle implies that the endpoint is part of the graph, while an open circle implies it is not. Get creative with your sketch, but ensure your graph accurately represents the intervals and formulas given.

Function Intervals

Understanding function intervals is key when dealing with piecewise functions. Each interval tells you where a particular equation applies. In this context, intervals can be found in inequalities, like "for \( x \leq -2 \)" or "\( -2 < x < 1 \)."
To handle these intervals, follow these easy steps:

• Read each segment's inequality correctly to identify whether endpoints are included.
• Note the transitions between pieces so you can join them cleanly on your graph.

Each interval's behavior may differ. For instance, in the current piecewise function:

• For the interval \( x \leq -2 \), the graph is linear, and the endpoint \( (-2) \) is included, indicated by a filled circle.
• In the interval \( -2 < x < 1 \), the graph forms a parabola, but does not include endpoints \( x = -2 \) or \( x = 1 \), shown by open circles.
• Finally, for \( x \geq 1 \), begin at \( x = 1 \) and extend a linear graph to the right, with a closed circle at \( x = 1 \).

These intervals thus guide you on how to draw and connect the function's segments accurately.

Parabolas

Parabolas are important curves in mathematics, representing quadratic functions. The function in this exercise, \( f(x) = -x^2 \), is an example of a downward-opening parabola. This shape is determined by the negative sign in front of \( x^2 \).
When sketching parabolas, it helps to know some key points:

• The vertex is the peak (or turning point); for \( f(x) = -x^2 \), the vertex is at the origin, \( (0, 0) \).
• Symmetry means the parabola is mirror-like about its vertical line, here coinciding with the y-axis.
• Check values at the endpoints of the interval to understand its engagement without strictly including them—for example, \( -1 \) and \( 1 \) in this function return \( -1 \) each.

Understanding these points helps not just in drawing the curve, but also in predicting the behavior across intervals. Keeping each segment's rules in mind prevents mixing up parabolic and linear behaviors in the sketch.