Graphs as Solution Sets, Linear-Quadratic Systems and Exponential Expressions: Algebra 1 Math Test Answer Key – PDF Download

Algebra 1 Graphs Test Answer Key: questions 1–5

Question 1: Points on a graph (2 points)

1. \((-3)^{2}-5=9-5=4\), so yes, \((-3,4)\) is on the graph.
2. \(2^{2}-5=-1\neq 1\), so no, \((2,1)\) is not on the graph.

Scoring: 1 point for each correct decision supported by substitution.

Question 2: Solutions on a line (2 points)

1. P and R lie on the line: \(-2(1)+3=1\) and \(-2(-1)+3=5\). P and R
2. For Q: \(-2(2)+3=-1\), not 0, so Q is not a solution. (For S: \(-2(0)+3=3\neq 2\).)

Scoring: 1 point for naming exactly P and R; 1 point for a correct substitution showing a mismatch.

Question 3: Line and parabola system (3 points)

\(x^{2}-6x+8=2x-7\) gives \(x^{2}-8x+15=0\), so \((x-3)(x-5)=0\) and \(x=3\) or \(x=5\). Then \(y=2(3)-7=-1\) and \(y=2(5)-7=3\).

Solutions: \((3,-1)\) and \((5,3)\). Check: \(9-18+8=-1\) and \(25-30+8=3\).

Scoring: 1 point for setting the expressions equal and reaching a correct equation equal to 0; 1 point for the x-values; 1 point for both y-values written as ordered pairs.

Question 4: Reading a system graph (3 points)

1. The graphs meet at \((-1,-3)\) and \((2,0)\).
2. \(x^{2}-4=x-2\) gives \(x^{2}-x-2=0\), so \((x-2)(x+1)=0\) and \(x=2\) or \(x=-1\). Then \(y=0\) and \(y=-3\), which matches the graph.

Scoring: 1 point for both points read from the graph; 1 point for the correct quadratic and factoring; 1 point for the y-values and agreement with the graph.

Question 5: How many solutions? (2 points)

1. \(x^{2}+1=1\) gives \(x=0\): one solution, \((0,1)\).
2. \(x^{2}+1=0\) gives \(x^{2}=-1\), which has no real solution: none.

Scoring: 1 point for each correct count with supporting algebra.

Algebra 1 Graphs Test Answer Key: questions 6–8

Question 6: Features of an exponential (3 points)

1. \(f(0)=4\cdot 1=4\), so \((0,4)\).
2. \(2^{x}\) is always positive, so \(4\cdot 2^{x}\) is never 0.
3. As \(x\) gets very large, \(f(x)\) gets very large; as \(x\) gets very small, \(f(x)\) approaches 0.

Scoring: 1 point per part; part (c) needs both directions.

Question 7: Rewrite exponentials (3 points)

(a) \(16^{x}=(2^{4})^{x}=2^{4x}\).

(b) \(1.07^{t}=(1.07^{1/12})^{12t}\approx(1.0057)^{12t}\), a monthly rate of about 0.57%.

Scoring: 1 point for (a); in (b), 1 point for the exponent rewrite and 1 point for the rate 0.57%.

Question 8: Find the exponential (2 points)

1. The y-intercept gives \(a=2\), and each step right multiplies \(y\) by 3, so \(y=2\cdot 3^{x}\).
2. As \(x\) gets very small, \(y\) approaches 0 (the x-axis is a horizontal asymptote).

Scoring: 1 point for the equation with both a and b correct; 1 point for the end behavior.

Scoring summary

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