Question 1: Identify the shifts (2 points)
(a) \(g(x)=f(x)-5\): the graph shifts down 5 units; the vertex is \((0,-5)\).
(b) \(h(x)=f(x+6)\): the graph shifts left 6 units; the vertex is \((-6,0)\).
Scoring: 1 point each for the correct direction and distance. Reversing left and right earns 0 for part (b).
Question 2: Equation from a graph (2 points)
The vertex is \((-2,3)\), so \(h=-2\) and \(k=3\). \(g(x)=(x+2)^{2}+3\)
Check with \((0,7)\): \((0+2)^{2}+3=7\).
Scoring: 1 point for \(h=-2\) (written as \(x+2\)); 1 point for \(k=3\).
Question 3: Combine transformations (2 points)
Stretch by 3 and flip: \(-3|x|\). Shift right 2: \(-3|x-2|\). Shift up 1: \(-3|x-2|+1\).
\(g(x)=-3|x-2|+1\), vertex \((2,1)\).
Scoring: 1 point for the factor \(-3\) and the form \(a|x-h|+k\); 1 point for \(h=2\) and \(k=1\) (the vertex).
Question 4: Graph an absolute value function (3 points)
(a) \(h=-1\), \(k=4\): the vertex is \((-1,4)\).
(b) \(-|x+1|+4=0\) gives \(|x+1|=4\), so \(x=3\) or \(x=-5\): \((3,0)\) and \((-5,0)\).
(c) Plot the vertex, the intercepts and \((0,3)\), \((-2,3)\), then draw the two rays.
Scoring: 1 point each for (a), (b) and (c); in (c) the graph must open down with the correct vertex and intercepts.
Question 5: Evaluate a piecewise function (2 points)
\(f(-2)=2(-2)+3=-1\) (since \(-2 < 1\)).
\(f(1)=8-1^{2}=7\) (since \(1\ge 1\)).
\(f(3)=8-3^{2}=8-9=-1\). \(-1\), \(7\), \(-1\)
Scoring: 2 points for all three correct; 1 point for exactly two correct.
Question 6: Graph a piecewise function (3 points)
Left piece: \(f(-2)=4\), \(f(0)=2\); at \(x=1\) the rule gives 1, an open circle at \((1,1)\).
Right piece: \(f(1)=2-3=-1\), a closed circle at \((1,-1)\); \(f(2)=1\), \(f(3)=3\).
Scoring: 1 point for the correct left piece; 1 point for the correct right piece; 1 point for the correct open and closed circles at \(x=1\).
Question 7: Read a step function (2 points)
(a) At \(x=2\) the closed circle is at height 6. \(\$6\)
(b) 2.5 pounds is between 2 and 3, where the height is 8. \(\$8\)
Scoring: 1 point for each correct cost. Answering \(\$7\) for part (b) by connecting the steps earns 0.
Question 8: Tolerance of rods (2 points)
(a) \(|11.7-12|=|-0.3|=0.3>0.25\). Not acceptable.
(b) \(-0.25\le x-12\le 0.25\), so \(11.75\le x\le 12.25\).
Scoring: 1 point for (a) with the value 0.3 compared with 0.25; 1 point for (b).
Question 9: Inside or outside (2 points)
\(y=(x+3)^{2}\) shifts \(y=x^{2}\) left 3: vertex \((-3,0)\); at \(x=0\), \(y=9\), so the y-intercept is \((0,9)\).
\(y=x^{2}+3\) shifts \(y=x^{2}\) up 3: vertex \((0,3)\); at \(x=0\), \(y=3\), so the y-intercept is \((0,3)\).
The vertices and y-intercepts differ, so the graphs are different.
Scoring: 1 point for the two correct vertices; 1 point for the two correct y-intercepts.
20 points
See every chapter on the Algebra 1 math page.