Graph this function.

Question

f(x)=
3x if x<1
–
7x+10 if x≥1
Select points on the graph to plot them. Select "Point fill" to change a point from closed to open.
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Answer 1

To graph the piecewise function

\[ f(x) = \begin{cases} 3x & \text{if } x < 1 \ -7x + 10 & \text{if } x \geq 1 \end{cases} \]

we can calculate a few points for each piece.

For \( x < 1 \) (using \( f(x) = 3x \)):

• At \( x = 0 \): \[ f(0) = 3(0) = 0 \quad \Rightarrow \quad (0, 0) \]
• At \( x = 0.5 \): \[ f(0.5) = 3(0.5) = 1.5 \quad \Rightarrow \quad (0.5, 1.5) \]
• At \( x = 0.9 \): \[ f(0.9) = 3(0.9) = 2.7 \quad \Rightarrow \quad (0.9, 2.7) \]

For \( x \geq 1 \) (using \( f(x) = -7x + 10 \)):

• At \( x = 1 \): \[ f(1) = -7(1) + 10 = 3 \quad \Rightarrow \quad (1, 3) \]
• At \( x = 2 \): \[ f(2) = -7(2) + 10 = -4 \quad \Rightarrow \quad (2, -4) \]
• At \( x = 3 \): \[ f(3) = -7(3) + 10 = -11 \quad \Rightarrow \quad (3, -11) \]

Summary of Points to Plot:

• For \( x < 1 \):

  • (0, 0) (closed point at \( x = 1 \))
  • (0.5, 1.5)
  • (0.9, 2.7)

• For \( x = 1 \):

  • (1, 3) (closed point at \( x = 1 \))

• For \( x \geq 1 \):

  • (2, -4)
  • (3, -11)

Plot Points with Consideration for Open and Closed Points:

• At \( x = 1 \), the point is closed because it appears in both conditions.
• For \( x < 1 \), the endpoint at \( (1, 3) \) is a closed point, while points for \( x < 1 \) are open.

Final Remark:

• The graph will feature a straight line for the first part starting from the origin to just before the point (1,3), and from (1,3) onwards, it will slope downwards from that point.

When graphing, make sure to clearly depict these two segments and indicate the closed point at \( x = 1 \).

Answer 2

Write the equation of the parabola that passes through the points (2,0), (
–
1,0), and (3,8).
Write your answer in the form y=a(x–p)(x–q), where a, p, and q are integers, decimals, or simplified fractions.