To graph the function \( f(x) = -7\sqrt{5}(x+9) + 12 \), we can break down the transformations involved in this equation.
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Horizontal Shift: The term \( (x + 9) \) indicates a horizontal shift. Specifically, \( f(x) \) is shifted 9 units to the left. This is because adding 9 to \( x \) means that the function will output the same value at \( x + 9 \) as it would have output at \( x \).
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Vertical Stretch and Reflection: The coefficient \( -7\sqrt{5} \) in front of the square root affects the vertical stretch and reflection. The negative sign indicates that the function is reflected over the x-axis. The magnitude \( 7\sqrt{5} \) indicates that it is vertically stretched by a factor of \( 7\sqrt{5} \).
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Vertical Shift: The \( +12 \) at the end of the function indicates a vertical shift upwards by 12 units.
Putting it all together, the transformations to graph \( f(x) = -7\sqrt{5}(x+9) + 12 \) are:
- Shift the graph left by 9 units.
- Reflect it over the x-axis.
- Stretch it vertically by a factor of \( 7\sqrt{5} \).
- Shift it upwards by 12 units.
This sequence of transformations will help you graph the function accurately.