Graphing Quadratic Functions: Algebra 1 Math Test Answer Key – PDF Download

Question 1: Features from standard form (2 points)

1. \(a=2 > 0\), so it opens up. The y-intercept is \(f(0)=6\): \((0, 6)\).
2. \(x=-\frac{-8}{2(2)}=2\) and \(f(2)=8-16+6=-2\). Axis \(x=2\), vertex \((2, -2)\).

Scoring: 1 point for direction and y-intercept (both needed); 1 point for the axis and the vertex.

Question 2: Read a graph (2 points)

1. Vertex \((1, 4)\); axis \(x=1\).
2. x-intercepts \((-1, 0)\) and \((3, 0)\); y-intercept \((0, 3)\).

Scoring: 1 point for the vertex and axis; 1 point for all three intercepts.

Question 3: Factored form (2 points)

1. Zeros: \(x=-1\) and \(x=5\). The axis is halfway: \(x=\frac{-1+5}{2}=\) 2.
2. \(g(2)=(3)(-3)=-9\). Vertex: \((2, -9)\).

Scoring: 1 point for both zeros and the axis; 1 point for the vertex.

Question 4: Complete the square (3 points)

\(h(x)=(x^{2}+10x+25)-25+16=\) \((x+5)^{2}-9\).

Vertex: \((-5, -9)\).

Zeros: \((x+5)^{2}=9\), so \(x+5=\pm 3\) and \(x=-2\) or \(x=-8\). Check: \((x+2)(x+8)=x^{2}+10x+16\).

Scoring: 1 point for the vertex form, 1 point for the vertex, 1 point for both zeros.

Question 5: Change the form (2 points)

1. \(-2(x^{2}-2x+1)+8=\) \(-2x^{2}+4x+6\).
2. Set \(y=0\): \((x-1)^{2}=4\), so \(x=3\) or \(x=-1\). Factored form: \(y=-2(x+1)(x-3)\). Check: \(-2(x^{2}-2x-3)=-2x^{2}+4x+6\).

Scoring: 1 point for correct expansion; 1 point for finding the zeros and keeping the factor \(-2\).

Question 6: Compare two functions (2 points)

1. The values repeat around \(x=2\) and the lowest is \(-2\). Vertex \((2, -2)\); axis \(x=2\).
2. For \(r\): \(x=-1\) and \(r(-1)=1-2-5=-6\). Since \(-6 < -2\), \(r\) has the smaller minimum.

Scoring: 1 point for the vertex and axis; 1 point for the minimum of \(r\) and the correct comparison.

Question 7: Kicked ball (3 points)

1. \(t=-\frac{48}{2(-16)}=1.5\) and \(h(1.5)=-36+72=36\). 36 feet at 1.5 seconds.
2. \(-16t(t-3)=0\), so \(t=0\) (the kick) or \(t=3\) seconds.
3. \(h(1)=-16+48=\) 32 feet.

Scoring: 1 point for each part (a), (b) and (c).

Question 8: Intervals of a function (2 points)

1. The vertex is \((-1, -4)\), so \(f\) is decreasing for \(x < -1\).
2. The graph is below the x-axis between the zeros \(-3\) and 1: \(-3 < x < 1\).

Scoring: 1 point for each part. Accept interval notation.

Question 9: Pen with fence (2 points)

1. The third side is \(60-2x\), so \(A(x)=x(60-2x)=-2x^{2}+60x\).
2. \(x=-\frac{60}{2(-2)}=15\) and \(A(15)=15(30)=\) 450 square feet.

Scoring: 1 point for the function; 1 point for finding the vertex and the area.

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