Problem 1: Shift down a parabola
\(g(x)=f(x)-6\), so every output decreases by 6. The graph shifts down 6 units.
The vertex of \(f\) is \((0,0)\), so the vertex of \(g\) is \((0,-6)\).
Problem 2: Shift up an absolute value
\(g(x)=f(x)+7\), so every output increases by 7. The graph shifts up 7 units.
The vertex moves from \((0,0)\) to \((0,7)\).
Problem 3: Equation from a shifted parabola
The vertex moved from \((0,0)\) to \((5,0)\), which is 5 units right. A move right by 5 uses \(x-5\) inside.
\(g(x)=f(x-5)\), so \(g(x)=(x-5)^{2}\).
Check with \((6,1)\): \((6-5)^{2}=1\). Check with \((3,4)\): \((3-5)^{2}=4\).
Problem 4: Shift left
\(g(x)=f(x+3)\), so the graph shifts left 3 units. The vertex is \((-3,0)\).
The y-intercept: \(g(0)=|0+3|=3\). The y-intercept is \((0,3)\).
Problem 5: Add outside or inside
\(f(3)+2=3^{2}+2=9+2=11\).
\(f(3+2)=f(5)=5^{2}=25\).
The results differ because the first adds 2 to the output (a vertical shift) and the second adds 2 to the input (a horizontal shift). 11 and 25
Problem 6: Stretch table
Substitute each value:
| \(x\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) |
|---|---|---|---|---|---|
| \(f(x)\) | 2 | 1 | 0 | 1 | 2 |
| \(g(x)\) | 6 | 3 | 0 | 3 | 6 |
Each y-value of \(g\) is 3 times the matching y-value of \(f\). The graph is a vertical stretch by a factor of 3; the vertex stays at \((0,0)\).
Problem 7: Equation from an absolute value graph
The vertex is \((2,-3)\), so \(h=2\) and \(k=-3\).
\(g(x)=|x-2|-3\)
Check with \((5,0)\): \(|5-2|-3=3-3=0\). Check with \((0,-1)\): \(|0-2|-3=2-3=-1\).
Problem 8: Reflect and shrink
Compare \((2,4)\) on \(f\) with \((2,-2)\) on \(g\): \(-2=-\frac{1}{2}\cdot 4\). Compare \(f(4)=16\) with \(g(4)=-8\): \(-8=-\frac{1}{2}\cdot 16\).
So \(g(x)=-\frac{1}{2}f(x)\), that is \(g(x)=-\frac{1}{2}x^{2}\).
The graph is flipped over the x-axis (the negative sign) and shrunk vertically by a factor of \(\frac{1}{2}\).
Problem 9: Transform a linear function
(a) \(g(x)=(2x+1)+3=2x+4\).
(b) \(h(x)=2(x+3)+1=2x+6+1=2x+7\).
(c) \(k(x)=-(2x+1)=-2x-1\).
Notice that shifting the line left 3 units gave the same line as shifting it up 6 units, because the slope is 2. (a) \(2x+4\); (b) \(2x+7\); (c) \(-2x-1\)
Problem 10: Vertex and range
Here \(h=4\) and \(k=2\). The vertex is \((4,2)\).
The axis of symmetry is the vertical line through the vertex: \(x=4\).
The graph opens up, so the minimum value is 2.
Domain: all real numbers. Range: \(y\ge 2\).
Problem 11: Graph an absolute value function
Substitute each \(x\):
| \(x\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) |
|---|---|---|---|---|---|---|---|
| \(y\) | 1 | 0 | -1 | -2 | -1 | 0 | 1 |
For example, \(x=-2\): \(|-2-1|-2=3-2=1\). Plot the points and draw two rays meeting at the vertex \((1,-2)\).
Problem 12: Intercepts of an absolute value graph
y-intercept: \(g(0)=2|0+1|-4=2-4=-2\), so \((0,-2)\).
x-intercepts: set \(2|x+1|-4=0\). Then \(2|x+1|=4\) and \(|x+1|=2\). So \(x+1=2\) or \(x+1=-2\), giving \(x=1\) or \(x=-3\).
x-intercepts \((1,0)\) and \((-3,0)\)
Problem 13: An absolute value that opens down
The vertex is \((3,5)\). Because \(a=-1 < 0\), the graph opens down and the maximum value is 5.
y-intercept: \(h(0)=-|0-3|+5=-3+5=2\), so \((0,2)\).
x-intercepts: \(-|x-3|+5=0\) gives \(|x-3|=5\), so \(x-3=5\) or \(x-3=-5\), and \(x=8\) or \(x=-2\). \((8,0)\) and \((-2,0)\)
Problem 14: Evaluate a piecewise function
\(-3 < 0\), so use \(x+4\): \(f(-3)=-3+4=1\).
\(-1 < 0\): \(f(-1)=-1+4=3\).
\(0\ge 0\), so use \(x^{2}\): \(f(0)=0^{2}=0\).
\(2\ge 0\): \(f(2)=2^{2}=4\). 1, 3, 0, 4
Problem 15: Graph a piecewise function
Left piece, \(x < 2\): \(f(-4)=-1\), \(f(0)=1\), and at \(x=2\) the rule gives \(\frac{1}{2}(2)+1=2\), an open circle at \((2,2)\).
Right piece, \(x\ge 2\): \(f(2)=-2+5=3\), a closed circle at \((2,3)\); then \(f(3)=2\) and \(f(6)=-1\).
Problem 16: Write a piecewise function
Piece 1: horizontal at \(y=2\), with a closed circle at \(x=-1\): \(f(x)=2\) for \(-4\le x\le -1\).
Piece 2: from \((-1,0)\) (open) to \((3,4)\) (closed), slope \(\frac{4-0}{3-(-1)}=1\), so \(y=x+1\) for \(-1 < x\le 3\).
Piece 3: from \((3,3)\) (open) to \((6,0)\), slope \(\frac{0-3}{6-3}=-1\), so \(y=6-x\) for \(3 < x\le 6\).
\(f(-4)=2\); \(f(3)=3+1=4\) (piece 2, because 3 is included there); \(f(5)=6-5=1\). 2, 4, 1
Problem 17: Greatest integer step function
\(s(3.7)=3\), because 3 is the greatest integer not exceeding 3.7.
\(s(-1.2)=-2\), because \(-2\le -1.2\) but \(-1>-1.2\).
\(s(5)=5\) and \(s(-4)=-4\), since integers map to themselves.
\(s(0.99)=0\). 3, -2, 5, -4, 0
Problem 18: Check a point
\(g(4)=-2|4-1|+6=-2(3)+6=0\). The y-value is 0, so \((4,0)\) is on the graph.
\(g(-2)=-2|-2-1|+6=-2(3)+6=0\), not 2. So \((-2,2)\) is not on the graph (the point \((-2,0)\) is).
Problem 19: Ride-share booking fee
Adding 4 to every output shifts the graph up 4 units. \(g(m)=2m+4\).
\(g(7)=2(7)+4=14+4=18\). The 7-mile ride costs \(\$18\).
Problem 20: Starting to save later
Everything happens 3 weeks later, so replace \(w\) with \(w-3\): \(g(w)=f(w-3)=15(w-3)\) for \(w\ge 3\). The graph shifts right 3 units.
\(g(8)=15(8-3)=15(5)=75\). \(\$75\)
Problem 21: Dropped ball
\(h(t)=-16f(t)+64\): stretch vertically by a factor of 16, flip over the t-axis, then shift up 64 units.
Starting height: \(h(0)=64\) feet.
Landing: \(-16t^{2}+64=0\), so \(t^{2}=4\) and \(t=2\) (time cannot be negative). The ball lands after 2 seconds.
Problem 22: Data plan
\(C(g)=20\) for \(0\le g\le 5\), and \(C(g)=20+4(g-5)\) for \(g>5\).
\(C(3)=20\). \(C(8)=20+4(3)=32\). \(C(12)=20+4(7)=48\).
\(\$20\), \(\$32\) and \(\$48\)
Problem 23: Parking garage
(a) 2.3 hours is in the interval from 2 to 3, where the height is 15. \(\$15\)
(b) At \(x=3\) the closed circle is at height 15. \(\$15\)
(c) Just past 3, the cost jumps to the next step. \(\$20\)
Check: \(5\times 3=15\) for 2.3 hours (3 started hours), and \(5\times 4=20\) for 3.01 hours.
Problem 24: Bottle tolerance
15.9 oz: \(E=|15.9-16|=|-0.1|=0.1\le 0.2\). Acceptable.
16.3 oz: \(E=|16.3-16|=0.3>0.2\). Not acceptable.
\(|x-16|\le 0.2\) means \(-0.2\le x-16\le 0.2\), so \(15.8\le x\le 16.2\).
See every chapter on the Algebra 1 math page.