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The graph of a piecewise function is shown.A graph has a straight line starting in quadrant 3 with a positive slope and right e...Asked by freedom
The graph of a piecewise function is shown.A graph has a straight line starting in quadrant 3 with a positive slope and right endpoint solid circle left-parenthesis negative 1 comma negative 1 right parenthesis is shown. It also shows a quadratic opening upwards from open circle left-parenthesis negative 1 comma negative 2 right-parenthesis to closed circle left-parenthesis 3 comma 2 right parenthesis. It also shows a quadratic with vertex left-parenthesis 5 comma 4 right-parenthesis opening downwards from left-parenthesis 3 comma 2 right-parenthesis to closed circle left-parenthesis 9 comma negative 4 right parenthesis.
Question
Use the graph to answer the question.
Write the equation of the piecewise function whose graph is shown.
(1 point)
Responses
math expressionf left parenthesis x right parenthesis equals enlarged left brace start layout 1st row start fraction 3 over 2 end fraction x plus one half comma x less than or equals negative 1 2nd row start fraction 1 over 4 end fraction left parenthesis x plus 1 right parenthesis squared minus 2 comma negative 1 less than x less than or equals 3 3rd row negative one half left parenthesis x minus 5 right parenthesis squared plus 4 comma 3 less than x less than or equals 9 end layout
Image with alt text: math expression f left parenthesis x right parenthesis equals enlarged left brace start layout 1st row start fraction 3 over 2 end fraction x plus one half comma x less than or equals negative 1 2nd row start fraction 1 over 4 end fraction left parenthesis x plus 1 right parenthesis squared minus 2 comma negative 1 less than x less than or equals 3 3rd row negative one half left parenthesis x minus 5 right parenthesis squared plus 4 comma 3 less than x less than or equals 9 end layout
math expressionf left parenthesis x right parenthesis equals enlarged left brace start layout 1st row start fraction 3 over 2 end fraction x plus one half comma x less than negative 1 2nd row start fraction 1 over 4 end fraction left parenthesis x plus 1 right parenthesis squared minus 2 comma negative 1 less than or equals x less than or equals 3 3rd row negative one half left parenthesis x minus 5 right parenthesis squared plus 4 comma 3 less than x less than or equals 9 end layout
Image with alt text: math expression f left parenthesis x right parenthesis equals enlarged left brace start layout 1st row start fraction 3 over 2 end fraction x plus one half comma x less than negative 1 2nd row start fraction 1 over 4 end fraction left parenthesis x plus 1 right parenthesis squared minus 2 comma negative 1 less than or equals x less than or equals 3 3rd row negative one half left parenthesis x minus 5 right parenthesis squared plus 4 comma 3 less than x less than or equals 9 end layout
math expressionf left parenthesis x right parenthesis equals enlarged left brace start layout 1st row start fraction 3 over 2 end fraction x plus one half comma x less than or equals negative 1 2nd row start fraction 1 over 4 end fraction left parenthesis x plus 1 right parenthesis squared minus 2 comma negative 1 less than x less than or equals 3 3rd row negative one half left parenthesis x minus 5 right parenthesis squared plus 4 comma 3 less than x less than or equals 9 end layout
Image with alt text: math expression f left parenthesis x right parenthesis equals enlarged left brace start layout 1st row start fraction 3 over 2 end fraction x plus one half comma x less than or equals negative 1 2nd row start fraction 1 over 4 end fraction left parenthesis x plus 1 right parenthesis squared minus 2 comma negative 1 less than x less than or equals 3 3rd row negative one half left parenthesis x minus 5 right parenthesis squared plus 4 comma 3 less than x less than or equals 9 end layout
math expressionf left parenthesis x right parenthesis equals enlarged left brace start layout 1st row start fraction 3 over 2 end fraction x plus one half comma x less than or equals negative 1 2nd row start fraction 1 over 4 end fraction left parenthesis x plus 1 right parenthesis squared minus 2 comma negative 1 less than x less than or equals 3 3rd row negative one half left parenthesis x minus 3 right parenthesis squared plus 4 comma 3 less than x less than or equals 9 end layout
Image with alt text: math expression f left parenthesis x right parenthesis equals enlarged left brace start layout 1st row start fraction 3 over 2 end fraction x plus one half comma x less than or equals negative 1 2nd row start fraction 1 over 4 end fraction left parenthesis x plus 1 right parenthesis squared minus 2 comma negative 1 less than x less than or equals 3 3rd row negative one half left parenthesis x minus 3 right parenthesis squared plus 4 comma 3 less than x less than or equals 9 end layout
Question
Use the graph to answer the question.
Write the equation of the piecewise function whose graph is shown.
(1 point)
Responses
math expressionf left parenthesis x right parenthesis equals enlarged left brace start layout 1st row start fraction 3 over 2 end fraction x plus one half comma x less than or equals negative 1 2nd row start fraction 1 over 4 end fraction left parenthesis x plus 1 right parenthesis squared minus 2 comma negative 1 less than x less than or equals 3 3rd row negative one half left parenthesis x minus 5 right parenthesis squared plus 4 comma 3 less than x less than or equals 9 end layout
Image with alt text: math expression f left parenthesis x right parenthesis equals enlarged left brace start layout 1st row start fraction 3 over 2 end fraction x plus one half comma x less than or equals negative 1 2nd row start fraction 1 over 4 end fraction left parenthesis x plus 1 right parenthesis squared minus 2 comma negative 1 less than x less than or equals 3 3rd row negative one half left parenthesis x minus 5 right parenthesis squared plus 4 comma 3 less than x less than or equals 9 end layout
math expressionf left parenthesis x right parenthesis equals enlarged left brace start layout 1st row start fraction 3 over 2 end fraction x plus one half comma x less than negative 1 2nd row start fraction 1 over 4 end fraction left parenthesis x plus 1 right parenthesis squared minus 2 comma negative 1 less than or equals x less than or equals 3 3rd row negative one half left parenthesis x minus 5 right parenthesis squared plus 4 comma 3 less than x less than or equals 9 end layout
Image with alt text: math expression f left parenthesis x right parenthesis equals enlarged left brace start layout 1st row start fraction 3 over 2 end fraction x plus one half comma x less than negative 1 2nd row start fraction 1 over 4 end fraction left parenthesis x plus 1 right parenthesis squared minus 2 comma negative 1 less than or equals x less than or equals 3 3rd row negative one half left parenthesis x minus 5 right parenthesis squared plus 4 comma 3 less than x less than or equals 9 end layout
math expressionf left parenthesis x right parenthesis equals enlarged left brace start layout 1st row start fraction 3 over 2 end fraction x plus one half comma x less than or equals negative 1 2nd row start fraction 1 over 4 end fraction left parenthesis x plus 1 right parenthesis squared minus 2 comma negative 1 less than x less than or equals 3 3rd row negative one half left parenthesis x minus 5 right parenthesis squared plus 4 comma 3 less than x less than or equals 9 end layout
Image with alt text: math expression f left parenthesis x right parenthesis equals enlarged left brace start layout 1st row start fraction 3 over 2 end fraction x plus one half comma x less than or equals negative 1 2nd row start fraction 1 over 4 end fraction left parenthesis x plus 1 right parenthesis squared minus 2 comma negative 1 less than x less than or equals 3 3rd row negative one half left parenthesis x minus 5 right parenthesis squared plus 4 comma 3 less than x less than or equals 9 end layout
math expressionf left parenthesis x right parenthesis equals enlarged left brace start layout 1st row start fraction 3 over 2 end fraction x plus one half comma x less than or equals negative 1 2nd row start fraction 1 over 4 end fraction left parenthesis x plus 1 right parenthesis squared minus 2 comma negative 1 less than x less than or equals 3 3rd row negative one half left parenthesis x minus 3 right parenthesis squared plus 4 comma 3 less than x less than or equals 9 end layout
Image with alt text: math expression f left parenthesis x right parenthesis equals enlarged left brace start layout 1st row start fraction 3 over 2 end fraction x plus one half comma x less than or equals negative 1 2nd row start fraction 1 over 4 end fraction left parenthesis x plus 1 right parenthesis squared minus 2 comma negative 1 less than x less than or equals 3 3rd row negative one half left parenthesis x minus 3 right parenthesis squared plus 4 comma 3 less than x less than or equals 9 end layout
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To write the piecewise function based on the description of the graph, we analyze each part of the graph and its corresponding mathematical representation:
1. **First segment:**
- A straight line that starts in quadrant 3 with a positive slope and has a solid endpoint at \((-1, -1)\).
- Since it is given that the endpoint is solid, the function value at \(x = -1\) would be included.
- The line can be calculated by finding its slope, which appears to be \(\frac{3}{2}\) (based on adjusting for the rise and run).
- Thus, the equation for this line is \(f(x) = \frac{3}{2}x + \frac{1}{2}\) for \(x \leq -1\).
2. **Second segment:**
- A quadratic function opening upwards from an open circle at \((-1, -2)\) to a closed circle at \((3, 2)\).
- The vertex form of a quadratic function can be expressed as \(f(x) = a(x-h)^2 + k\), where \((h, k)\) is the vertex.
- This equation has been identified as \(f(x) = \frac{1}{4}(x + 1)^2 - 2\) for \(-1 < x \leq 3\).
3. **Third segment:**
- A quadratic function opening downwards with a vertex at \((5, 4)\) starting from the closed endpoint at \((3, 2)\) to another closed endpoint at \((9, -4)\).
- The vertex form is \(f(x) = -\frac{1}{2}(x - 5)^2 + 4\) for \(3 < x \leq 9\).
Now we can summarize this in a piecewise function as follows:
\[
f(x) =
\begin{cases}
\frac{3}{2}x + \frac{1}{2}, & x \leq -1 \\
\frac{1}{4}(x + 1)^2 - 2, & -1 < x \leq 3 \\
-\frac{1}{2}(x - 5)^2 + 4, & 3 < x \leq 9
\end{cases}
\]
This is the mathematical representation of the piecewise function based on the given description of the graph.
1. **First segment:**
- A straight line that starts in quadrant 3 with a positive slope and has a solid endpoint at \((-1, -1)\).
- Since it is given that the endpoint is solid, the function value at \(x = -1\) would be included.
- The line can be calculated by finding its slope, which appears to be \(\frac{3}{2}\) (based on adjusting for the rise and run).
- Thus, the equation for this line is \(f(x) = \frac{3}{2}x + \frac{1}{2}\) for \(x \leq -1\).
2. **Second segment:**
- A quadratic function opening upwards from an open circle at \((-1, -2)\) to a closed circle at \((3, 2)\).
- The vertex form of a quadratic function can be expressed as \(f(x) = a(x-h)^2 + k\), where \((h, k)\) is the vertex.
- This equation has been identified as \(f(x) = \frac{1}{4}(x + 1)^2 - 2\) for \(-1 < x \leq 3\).
3. **Third segment:**
- A quadratic function opening downwards with a vertex at \((5, 4)\) starting from the closed endpoint at \((3, 2)\) to another closed endpoint at \((9, -4)\).
- The vertex form is \(f(x) = -\frac{1}{2}(x - 5)^2 + 4\) for \(3 < x \leq 9\).
Now we can summarize this in a piecewise function as follows:
\[
f(x) =
\begin{cases}
\frac{3}{2}x + \frac{1}{2}, & x \leq -1 \\
\frac{1}{4}(x + 1)^2 - 2, & -1 < x \leq 3 \\
-\frac{1}{2}(x - 5)^2 + 4, & 3 < x \leq 9
\end{cases}
\]
This is the mathematical representation of the piecewise function based on the given description of the graph.
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