Question

1. a) graph the function $f(x) = 3(x - 1)^2 - 4$. b) what does $f(-1)$ represent on the graph? indicate on the graph how you would find $f(-1)$. c) use the equation to determine i) $f(2) - f(1)$ ii) $2f(3) - 7$ iii) $f(1 - x)$

Explanation:

Part a) Graphing \( f(x) = 3(x - 1)^2 - 4 \)

This is a quadratic function in vertex form \( f(x) = a(x - h)^2 + k \), where:

• Vertex: \( (h, k) = (1, -4) \)
• Since \( a = 3 > 0 \), the parabola opens upward.
• To find additional points:
• When \( x = 0 \): \( f(0) = 3(0 - 1)^2 - 4 = 3(1) - 4 = -1 \), so the point is \( (0, -1) \).
• When \( x = 2 \): \( f(2) = 3(2 - 1)^2 - 4 = 3(1) - 4 = -1 \), so the point is \( (2, -1) \).
• When \( x = 3 \): \( f(3) = 3(3 - 1)^2 - 4 = 3(4) - 4 = 8 \), so the point is \( (3, 8) \).
• When \( x = -1 \): \( f(-1) = 3(-1 - 1)^2 - 4 = 3(4) - 4 = 8 \), so the point is \( (-1, 8) \).

Plot the vertex \( (1, -4) \), the symmetric points \( (0, -1) \), \( (2, -1) \), \( (-1, 8) \), \( (3, 8) \), and draw a smooth upward - opening parabola through these points.

Part b) Meaning of \( f(-1) \)

• \( f(-1) \) represents the \( y \) - value of the function when \( x=-1 \). Geometrically, it is the \( y \) - coordinate of the point on the graph of \( y = f(x) \) where \( x=-1 \).
• To find \( f(-1) \) on the graph: Locate \( x = -1 \) on the \( x \) - axis. Then, move vertically until you intersect the graph of \( y = f(x) \). The \( y \) - coordinate of this intersection point is \( f(-1) \). Using the equation \( f(-1)=3(-1 - 1)^2-4=3\times4 - 4 = 8 \), so the point is \( (-1, 8) \).

Part c)

i) Calculate \( f(2)-f(1) \)

• First, find \( f(2) \):
• Substitute \( x = 2 \) into \( f(x)=3(x - 1)^2-4 \).
• \( f(2)=3(2 - 1)^2-4=3\times1 - 4=-1 \)
• Then, find \( f(1) \):
• Substitute \( x = 1 \) into \( f(x)=3(x - 1)^2-4 \).
• \( f(1)=3(1 - 1)^2-4=3\times0 - 4=-4 \)
• Now, calculate \( f(2)-f(1) \):
• \( f(2)-f(1)=-1-(-4)=-1 + 4 = 3 \)

ii) Calculate \( 2f(3)-7 \)

• First, find \( f(3) \):
• Substitute \( x = 3 \) into \( f(x)=3(x - 1)^2-4 \).
• \( f(3)=3(3 - 1)^2-4=3\times4 - 4 = 8 \)
• Then, calculate \( 2f(3)-7 \):
• \( 2f(3)-7=2\times8 - 7=16 - 7 = 9 \)

iii) Calculate \( f(1 - x) \)

• Substitute \( x\) with \( 1 - x\) in \( f(x)=3(x - 1)^2-4 \).
• \( f(1 - x)=3((1 - x)-1)^2-4 \)
• Simplify the expression inside the parentheses: \( (1 - x)-1=-x \)
• Then \( f(1 - x)=3(-x)^2-4=3x^2-4 \)

Final Answers

• a) Graph with vertex \( (1, -4) \), opening upward, passing through points like \( (0, -1) \), \( (2, -1) \), \( (-1, 8) \), \( (3, 8) \)
• b) \( f(-1) \) is the \( y \) - value at \( x = -1 \), found by vertical intersection at \( x=-1 \) (point \( (-1, 8) \))
• c)
• i) \( \boldsymbol{3} \)
• ii) \( \boldsymbol{9} \)
• iii) \( \boldsymbol{3x^2-4} \)

Answer:

Part a) Graphing \( f(x) = 3(x - 1)^2 - 4 \)

This is a quadratic function in vertex form \( f(x) = a(x - h)^2 + k \), where:

• Vertex: \( (h, k) = (1, -4) \)
• Since \( a = 3 > 0 \), the parabola opens upward.
• To find additional points:
• When \( x = 0 \): \( f(0) = 3(0 - 1)^2 - 4 = 3(1) - 4 = -1 \), so the point is \( (0, -1) \).
• When \( x = 2 \): \( f(2) = 3(2 - 1)^2 - 4 = 3(1) - 4 = -1 \), so the point is \( (2, -1) \).
• When \( x = 3 \): \( f(3) = 3(3 - 1)^2 - 4 = 3(4) - 4 = 8 \), so the point is \( (3, 8) \).
• When \( x = -1 \): \( f(-1) = 3(-1 - 1)^2 - 4 = 3(4) - 4 = 8 \), so the point is \( (-1, 8) \).

Plot the vertex \( (1, -4) \), the symmetric points \( (0, -1) \), \( (2, -1) \), \( (-1, 8) \), \( (3, 8) \), and draw a smooth upward - opening parabola through these points.

Part b) Meaning of \( f(-1) \)

• \( f(-1) \) represents the \( y \) - value of the function when \( x=-1 \). Geometrically, it is the \( y \) - coordinate of the point on the graph of \( y = f(x) \) where \( x=-1 \).
• To find \( f(-1) \) on the graph: Locate \( x = -1 \) on the \( x \) - axis. Then, move vertically until you intersect the graph of \( y = f(x) \). The \( y \) - coordinate of this intersection point is \( f(-1) \). Using the equation \( f(-1)=3(-1 - 1)^2-4=3\times4 - 4 = 8 \), so the point is \( (-1, 8) \).

Part c)

i) Calculate \( f(2)-f(1) \)

• First, find \( f(2) \):
• Substitute \( x = 2 \) into \( f(x)=3(x - 1)^2-4 \).
• \( f(2)=3(2 - 1)^2-4=3\times1 - 4=-1 \)
• Then, find \( f(1) \):
• Substitute \( x = 1 \) into \( f(x)=3(x - 1)^2-4 \).
• \( f(1)=3(1 - 1)^2-4=3\times0 - 4=-4 \)
• Now, calculate \( f(2)-f(1) \):
• \( f(2)-f(1)=-1-(-4)=-1 + 4 = 3 \)

ii) Calculate \( 2f(3)-7 \)

• First, find \( f(3) \):
• Substitute \( x = 3 \) into \( f(x)=3(x - 1)^2-4 \).
• \( f(3)=3(3 - 1)^2-4=3\times4 - 4 = 8 \)
• Then, calculate \( 2f(3)-7 \):
• \( 2f(3)-7=2\times8 - 7=16 - 7 = 9 \)

iii) Calculate \( f(1 - x) \)

• Substitute \( x\) with \( 1 - x\) in \( f(x)=3(x - 1)^2-4 \).
• \( f(1 - x)=3((1 - x)-1)^2-4 \)
• Simplify the expression inside the parentheses: \( (1 - x)-1=-x \)
• Then \( f(1 - x)=3(-x)^2-4=3x^2-4 \)

Final Answers

• a) Graph with vertex \( (1, -4) \), opening upward, passing through points like \( (0, -1) \), \( (2, -1) \), \( (-1, 8) \), \( (3, 8) \)
• b) \( f(-1) \) is the \( y \) - value at \( x = -1 \), found by vertical intersection at \( x=-1 \) (point \( (-1, 8) \))
• c)
• i) \( \boldsymbol{3} \)
• ii) \( \boldsymbol{9} \)
• iii) \( \boldsymbol{3x^2-4} \)