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

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
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Answer 1

Based on the description of the piecewise function and the characteristics mentioned, we can determine the equations for each segment of the piecewise function:

1. For the linear segment (starting in quadrant 3 with a positive slope and ending at \((-1, -1)\)):

   • Since it starts from below and ends at \((-1, -1)\), we can represent this line as \(f(x) = \frac{3}{2}x + \frac{1}{2}\) for \(x \leq -1\).

2. For the quadratic segment (opening upwards from \((-1, -2)\) to \((3, 2)\)):

   • The vertex form of a quadratic function that has a minimum is used here. It can be expressed as: \[ f(x) = \frac{1}{4}(x + 1)^2 - 2 \] This applies for \(-1 < x \leq 3\).

3. For the quadratic segment (opening downwards from \((3, 2)\) to \((9, -4)\)):

   • The vertex is at \((5, 4)\) and opens downward. The vertex form would be: \[ f(x) = -\frac{1}{2}(x - 5)^2 + 4 \] This applies for \(3 < x \leq 9\).

Combining all parts together, we can write the piecewise function as:

\[ f(x) = \begin{cases} \frac{3}{2}x + \frac{1}{2} & \text{if } x \leq -1 \ \frac{1}{4}(x + 1)^2 - 2 & \text{if } -1 < x \leq 3 \ -\frac{1}{2}(x - 5)^2 + 4 & \text{if } 3 < x \leq 9 \end{cases} \]

This confirms the final answer.