Free Math Worksheets

- What does a point on the \(y\)-axis tell you about \(c\)? - Substitute each point into \(f(x) = ax^2 + bx + c\). - How can you solve the resulting system of equations? - Check the final function using all three points.

1. Substitute \(A(0, 4)\): \(c = 4\). 2. Substitute \(B(1, 3)\): \(a + b + 4 = 3\), so \(a + b = -1\). 3. Substitute \(C(2, 6)\): \(4a + 2b + 4 = 6\), so \(2a + b = 1\). 4. Subtract \(a + b = -1\) from \(2a + b = 1\): \(a = 2\). Then \(2 + b = -1\), so \(b = -3\). 5. Therefore, \(f(x) = 2x^2 - 3x + 4\).

\(a = 2\), \(b = -3\), \(c = 4\); \(f(x) = 2x^2 - 3x + 4\)

- Write one equation for each point. - Which pair of equations makes one variable easy to eliminate? - After finding one coefficient, reduce the system to two unknowns. - Keep track of each intermediate value for \(a\), \(b\), and \(c\).

1. Substitute the three points: \(a + b + c = 1\), \(4a + 2b + c = 0\), \(a - b + c = 9\). 2. Subtract the third equation from the first: \(2b = -8\), so \(b = -4\). 3. Substitute \(b = -4\) into the first two equations: \(a + c = 5\) and \(4a + c = 8\). 4. Subtract the equations: \(3a = 3\), so \(a = 1\). Then \(c = 4\). 5. Therefore, \(f(x) = x^2 - 4x + 4\).

\(f(x) = x^2 - 4x + 4\)

- Substitute each point into \(f(x) = ax^2 + bx + c\). - How many equations and unknowns do you have? - Which pair of equations makes one variable easy to eliminate? - Which elimination or substitution steps are most efficient?

1. Substitute the three points: \(a + b + c = 0\), \(4a + 2b + c = 1\), \(a - b + c = 10\). 2. Subtract the third equation from the first: \(2b = -10\), so \(b = -5\). 3. Substitute \(b = -5\) into the first two equations: \(a + c = 5\) and \(4a + c = 11\). 4. Subtract the equations: \(3a = 6\), so \(a = 2\). Then \(c = 3\). 5. Therefore, \(f(x) = 2x^2 - 5x + 3\).

\(f(x) = 2x^2 - 5x + 3\)

- What does the point \(A(0, 5)\) tell you immediately about \(c\)? - Substitute the known value of \(c\) into the other two point equations. - How do you test whether a point lies on a graph?

1. Substituting \(A(0, 5)\) gives \(c = 5\). 2. Substitute \(B(2, 1)\): \(4a + 2b + 5 = 1\), so \(2a + b = -2\). 3. Substitute \(C(4, 5)\): \(16a + 4b + 5 = 5\), so \(4a + b = 0\). 4. Subtract the equations: \(2a = 2\), so \(a = 1\). Then \(4 + b = 0\), so \(b = -4\). 5. Thus, \(f(x) = x^2 - 4x + 5\). 6. Evaluate \(f(5) = 5^2 - 4 \cdot 5 + 5 = 10\). Since this matches the point's \(y\)-coordinate, \(D(5, 10)\) lies on the graph.

\(f(x) = x^2 - 4x + 5\). Yes, \(D(5, 10)\) lies on the graph.

- What do you notice about the \(y\)-coordinates of \(A\) and \(C\)? - Where is the axis of symmetry between two points at the same height? - Which form is easiest when the vertex is known?

1. Points \(A\) and \(C\) have the same \(y\)-coordinate, so the axis of symmetry is halfway between their \(x\)-coordinates: \(x = \frac{1 + 5}{2} = 3\). 2. Since \(B(3, 3)\) lies on that axis, it is the vertex. Vertex form is therefore the most efficient choice: \(y = a(x - 3)^2 + 3\). 3. Substitute \(A(1, 7)\): \(7 = a(1 - 3)^2 + 3\), so \(4a = 4\) and \(a = 1\). 4. The function is \(y = (x - 3)^2 + 3\), or in standard form, \(y = x^2 - 6x + 12\).

Vertex form is the most efficient starting point. \(y = (x - 3)^2 + 3\), or \(y = x^2 - 6x + 12\) in standard form.

- When no zeros or vertex are given, standard form is a useful starting point. - Substitute each point to create three equations. - Eliminate one variable at a time from the system.

1. Use standard form \(y = ax^2 + bx + c\) and substitute the points: \(a - b + c = 11\), \(a + b + c = 3\), \(4a + 2b + c = 5\). 2. Subtract the first equation from the second: \(2b = -8\), so \(b = -4\). 3. Substitute \(b = -4\): the second equation gives \(a + c = 7\), and the third gives \(4a + c = 13\). 4. Subtract the equations: \(3a = 6\), so \(a = 2\). Then \(c = 5\). 5. Therefore, \(y = 2x^2 - 4x + 5\).

\(y = 2x^2 - 4x + 5\)

- What does a point with \(x = 0\) tell you about \(c\)? - Which coefficient determines a parabola's width and opening direction? - Do the points show a symmetry that helps locate the vertex?

1. Substituting \(P(0, 3)\) gives \(c = 3\). 2. Substitute \(Q(1, 2)\): \(a + b + 3 = 2\), so \(a + b = -1\). 3. Substitute \(R(2, 3)\): \(4a + 2b + 3 = 3\), so \(2a + b = 0\). 4. Subtract the equations: \(a = 1\), and then \(b = -2\). Thus, \(f(x) = x^2 - 2x + 3\). 5. Since \(|a| = 1\), the parabola has the same width as \(y = x^2\). 6. Rewrite \(f(x) = (x - 1)^2 + 2\), so the vertex is \((1, 2)\).

a) \(f(x) = x^2 - 2x + 3\) b) It has the same width as \(y = x^2\) because \(|a| = 1\). c) \((1, 2)\)

- What do you notice about the \(y\)-coordinates of \(A\) and \(C\)? - Which point lies on the axis of symmetry? - Which form is easiest when the vertex is known?

1. Points \(A\) and \(C\) have the same \(y\)-coordinate and are equally far from \(x = 0\), so the axis of symmetry is \(x = 0\). 2. Since \(B(0, 5)\) lies on the axis, it is the vertex. Vertex form is therefore the most efficient starting point: \(y = ax^2 + 5\). 3. Substitute \(C(2, 1)\): \(1 = 4a + 5\), so \(a = -1\). 4. Thus, the equation is \(y = -x^2 + 5\).

Vertex form is the best starting point, and the equation is \(y = -x^2 + 5\).

- Which point gives the \(y\)-intercept directly? - Substitute the other two points to create a system in \(a\) and \(b\). - How can you eliminate one variable from the two equations?

1. Substitute \(P_1(0, 1)\): \(c = 1\). 2. Substitute \(P_2(2, 5)\): \(4a + 2b + 1 = 5\), so \(2a + b = 2\). 3. Substitute \(P_3(4, 5)\): \(16a + 4b + 1 = 5\), so \(4a + b = 1\). 4. Subtract the equations: \(2a = -1\), so \(a = -0.5\). 5. Substitute into \(2a + b = 2\): \(-1 + b = 2\), so \(b = 3\). 6. Therefore, \(f(x) = -0.5x^2 + 3x + 1\).

\(f(x) = -0.5x^2 + 3x + 1\)

- What does \(P(0, 4)\) tell you about \(c\)? - What do you notice about the \(y\)-coordinates of \(P\) and \(R\)? - How does that symmetry identify the vertex?

1. Substituting \(P(0, 4)\) gives \(c = 4\). 2. Points \(P\) and \(R\) have the same \(y\)-coordinate, so the axis of symmetry is halfway between \(x = 0\) and \(x = 4\), at \(x = 2\). Thus, \(Q(2, 0)\) is the vertex. 3. Write vertex form as \(f(x) = a(x - 2)^2\). Substitute \(P(0, 4)\): \(4 = a(0 - 2)^2 = 4a\), so \(a = 1\). 4. Expand: \(f(x) = (x - 2)^2 = x^2 - 4x + 4\). Therefore, \(a = 1\), \(b = -4\), and \(c = 4\).

\(a = 1\), \(b = -4\), \(c = 4\); \(f(x) = x^2 - 4x + 4\)

- What does a \(y\)-intercept of \(5\) tell you about \(c\)? - How can the two additional points create two equations in \(a\) and \(b\)? - Which system-solving method is most efficient here?

1. A \(y\)-intercept of \(5\) gives \(c = 5\). 2. Substitute \(D(-1, 12)\): \(a - b + 5 = 12\), so \(a - b = 7\). 3. Substitute \(E(2, 9)\): \(4a + 2b + 5 = 9\), so \(2a + b = 2\). 4. Add the equations: \(3a = 9\), so \(a = 3\). 5. Substitute into \(a - b = 7\): \(3 - b = 7\), so \(b = -4\). 6. Therefore, \(p(x) = 3x^2 - 4x + 5\).

\(p(x) = 3x^2 - 4x + 5\)

- How do you compare slopes to test collinearity? - What value can you read directly from a point on the \(y\)-axis? - How can you solve the resulting two equations in \(a\) and \(b\)?

1. Compute two slopes: \(m_{AB} = \frac{2 - 10}{0 - (-2)} = -4\) and \(m_{BC} = \frac{2 - 2}{2 - 0} = 0\). Since the slopes are different, the points are not collinear. 2. Substituting \(B(0, 2)\) gives \(c = 2\). 3. Substitute \(A(-2, 10)\): \(4a - 2b + 2 = 10\), so \(4a - 2b = 8\). 4. Substitute \(C(2, 2)\): \(4a + 2b + 2 = 2\), so \(4a + 2b = 0\). 5. Add the equations: \(8a = 8\), so \(a = 1\). Then \(4 + 2b = 0\), so \(b = -2\). 6. Therefore, \(f(x) = x^2 - 2x + 2\).

The points are not collinear. The quadratic function is \(f(x) = x^2 - 2x + 2\).

- Compare the slopes from \(P\) to \(Q\) and from \(Q\) to \(R\). - Substitute each point into \(f(x) = ax^2 + bx + c\). - How can subtracting consecutive equations eliminate \(c\)?

1. The slopes are \(m_{PQ} = \frac{3 - 1}{2 - 1} = 2\) and \(m_{QR} = \frac{7 - 3}{3 - 2} = 4\). Since the slopes are different, the points are not collinear. 2. Substitute the points: \(a + b + c = 1\), \(4a + 2b + c = 3\), \(9a + 3b + c = 7\). 3. Subtract consecutive equations: \(3a + b = 2\) and \(5a + b = 4\). 4. Subtract these equations: \(2a = 2\), so \(a = 1\). Then \(3 + b = 2\), so \(b = -1\). 5. Substitute into \(a + b + c = 1\): \(c = 1\). 6. Therefore, \(f(x) = x^2 - x + 1\).

The points are not collinear. The quadratic function is \(f(x) = x^2 - x + 1\).

- Substitute each point into standard form. - How can subtracting equations eliminate \(c\)? - Solve the resulting two-equation system before finding the final coefficient.

1. Substitute the points: \(4a - 2b + c = 10\), \(a + b + c = -2\), \(9a + 3b + c = 0\). 2. Subtract the second equation from the first: \(3a - 3b = 12\), so \(a - b = 4\). 3. Subtract the second equation from the third: \(8a + 2b = 2\), so \(4a + b = 1\). 4. Add the two reduced equations: \(5a = 5\), so \(a = 1\). Then \(1 - b = 4\), so \(b = -3\). 5. Substitute into \(a + b + c = -2\): \(c = 0\). 6. Therefore, \(f(x) = x^2 - 3x\).

\(f(x) = x^2 - 3x\)

- What happens to the \(ax^2\) and \(bx\) terms when \(x = 0\)? - Which coefficient can you determine immediately from \(R\)? - Can adding or subtracting the remaining equations eliminate a variable?

1. Substitute the points: \(a + b + c = 5\), \(a - b + c = 1\), \(c = 2\). 2. Substitute \(c = 2\) into the first two equations: \(a + b = 3\) and \(a - b = -1\). 3. Add the equations: \(2a = 2\), so \(a = 1\). 4. Substitute into \(a + b = 3\): \(b = 2\). 5. Therefore, \(y = x^2 + 2x + 2\).

\(y = x^2 + 2x + 2\)

- Be careful with signs when substituting a negative \(x\)-value. - Which coefficients match in the equations from \(A\) and \(C\)? - Which elimination step removes two variables at once?

1. Substituting the points gives: \(4a - 2b + c = 10\), \(a + b + c = 1\), \(4a + 2b + c = 6\). 2. Subtract the equation from \(A\) from the equation from \(C\): \(4b = -4\), so \(b = -1\). 3. Substitute \(b = -1\) into the equations from \(B\) and \(C\): \(a + c = 2\) and \(4a + c = 8\). 4. Subtract: \(3a = 6\), so \(a = 2\). Then \(c = 0\). 5. Therefore, \(y = 2x^2 - x\).

a) \(4a - 2b + c = 10\), \(a + b + c = 1\), and \(4a + 2b + c = 6\) b) Subtracting the equations from \(A\) and \(C\) gives \(4b = -4\), so \(b = -1\). c) \(y = 2x^2 - x\)

- What does \(c\) represent on the graph? - Can you simplify either equation before combining them? - Check the final equation with both \(T\) and \(U\).

1. Since \(S\) has \(x = 0\), substituting it immediately gives \(c = 4\). 2. Substitute \(T(1, 3)\): \(a + b + 4 = 3\), so \(a + b = -1\). 3. Substitute \(U(-2, -6)\): \(4a - 2b + 4 = -6\), so \(2a - b = -5\). 4. Add the two equations: \(3a = -6\), so \(a = -2\). 5. Substitute into \(a + b = -1\): \(b = 1\). 6. Therefore, \(a = -2\), \(b = 1\), \(c = 4\), and \(y = -2x^2 + x + 4\).

Point \(S(0, 4)\) gives \(c = 4\) immediately. The coefficients are \(a = -2\), \(b = 1\), and \(c = 4\), so \(y = -2x^2 + x + 4\).

- Divide two output equations to eliminate \(a\). - Compare the ratio of outputs with a power of the ratio of inputs. - After finding \(n\), substitute one point to find \(a\). - Verify the equation with the remaining point.

1. Divide the values at \(x=3\) and \(x=2\): \(\frac{22.5}{10}=2.25\). From the model, this ratio equals \(\left(\frac{3}{2}\right)^n\). 2. Since \(\left(\frac{3}{2}\right)^2=2.25\), the exponent is \(n=2\). 3. Substitute \((2, 10)\): \(10=a\cdot 2^2\), so \(a=2.5\). 4. Check the third point: \(2.5\cdot 5^2=62.5\). Therefore, the equation fits all three points.

\(g(x)=2.5x^2\)

- Which point gives \(c\) immediately? - Use the other two points to create a system for \(a\) and \(b\). - After finding standard form, how can completing the square produce vertex form? - What information becomes visible in vertex form?

1. Substituting \(A(0, -1)\) gives \(c = -1\). 2. Substitute \(B(2, 5)\): \(4a + 2b - 1 = 5\), so \(4a + 2b = 6\). 3. Substitute \(C(-2, -3)\): \(4a - 2b - 1 = -3\), so \(4a - 2b = -2\). 4. Add the two equations: \(8a = 4\), so \(a = 0.5\). Then \(2 + 2b = 6\), so \(b = 2\). 5. The standard form is \(f(x) = 0.5x^2 + 2x - 1\). 6. Complete the square: \(f(x) = 0.5(x^2 + 4x) - 1 = 0.5((x + 2)^2 - 4) - 1 = 0.5(x + 2)^2 - 3\).

Standard form: \(f(x) = 0.5x^2 + 2x - 1\) Vertex form: \(f(x) = 0.5(x + 2)^2 - 3\)

- Write one equation for each of the three points. - Compare the equations from \(P\) and \(Q\). Which variable is easy to isolate? - After finding \(a\), \(b\), and \(c\), how can you rewrite the function to reveal the vertex?

1. Substitute the points: \(a - b + c = -5\), \(a + b + c = 3\), \(4a + 2b + c = 1\). 2. Subtract the first equation from the second: \(2b = 8\), so \(b = 4\). 3. Substitute \(b = 4\): the second equation gives \(a + c = -1\), and the third gives \(4a + c = -7\). 4. Subtract the equations: \(3a = -6\), so \(a = -2\). Then \(c = 1\). 5. The standard form is \(f(x) = -2x^2 + 4x + 1\). 6. Complete the square: \(f(x) = -2(x^2 - 2x) + 1 = -2((x - 1)^2 - 1) + 1 = -2(x - 1)^2 + 3\). The vertex is \((1, 3)\).

Standard form: \(f(x) = -2x^2 + 4x + 1\) Vertex form: \(f(x) = -2(x - 1)^2 + 3\) Vertex: \((1, 3)\)

- Use elimination to remove variables one at a time. - How can the equations from \(x = -1\) and \(x = 1\) simplify the system? - When completing the square with an odd linear coefficient, what number should you add and subtract?

1. Substitute the points: \(a - b + c = 1\), \(a + b + c = -1\), \(4a + 2b + c = 1\). 2. Subtract the second equation from the first: \(-2b = 2\), so \(b = -1\). 3. Substitute \(b = -1\). The second equation gives \(a + c = 0\), so \(c = -a\). The third gives \(4a - 2 + c = 1\). Thus, \(3a = 3\), so \(a = 1\) and \(c = -1\). 4. The standard form is \(f(x) = x^2 - x - 1\). 5. Complete the square: \(f(x) = x^2 - x + 0.25 - 0.25 - 1 = (x - 0.5)^2 - 1.25\).

Standard form: \(f(x) = x^2 - x - 1\) Vertex form: \(f(x) = (x - 0.5)^2 - 1.25\)

- Write one equation for each point. - Which subtraction quickly eliminates two variables? - After finding standard form, factor the leading coefficient before completing the square. - What number completes the square inside the parentheses?

1. Substitute the points: \(a - b + c = 6\), \(a + b + c = -2\), \(4a + 2b + c = 0\). 2. Subtract the first equation from the second: \(2b = -8\), so \(b = -4\). 3. Substitute \(b = -4\): the second equation gives \(a + c = 2\), and the third gives \(4a + c = 8\). 4. Subtract the equations: \(3a = 6\), so \(a = 2\). Then \(c = 0\). 5. The standard form is \(f(x) = 2x^2 - 4x\). 6. Complete the square: \(f(x) = 2(x^2 - 2x) = 2((x - 1)^2 - 1) = 2(x - 1)^2 - 2\).

a) \(f(x) = 2x^2 - 4x\) b) \(f(x) = 2(x - 1)^2 - 2\)

- Two points have \(y = 0\). What does that tell you about the graph? - Which subtraction simplifies the three-equation system? - How do you test whether a point lies on a graph? - How can completing the square produce vertex form?

1. Substitute the three points: \(9a - 3b + c = 0\), \(a + b + c = 0\), \(4a + 2b + c = 10\). 2. Subtract the second equation from the first: \(8a - 4b = 0\), so \(b = 2a\). 3. Subtract the second equation from the third: \(3a + b = 10\). Substitute \(b = 2a\): \(5a = 10\), so \(a = 2\) and \(b = 4\). 4. Substitute into \(a + b + c = 0\): \(c = -6\). Thus, \(f(x) = 2x^2 + 4x - 6\). 5. Complete the square: \(f(x) = 2(x^2 + 2x) - 6 = 2((x + 1)^2 - 1) - 6 = 2(x + 1)^2 - 8\). 6. Evaluate \(f(-2) = 2 \cdot (-2)^2 + 4 \cdot (-2) - 6 = -6\). Therefore, \(D(-2, -6)\) lies on the graph.

a) \(f(x) = 2x^2 + 4x - 6\) b) \(f(x) = 2(x + 1)^2 - 8\) c) Yes, \(D(-2, -6)\) lies on the graph.

- Compare the slopes between consecutive pairs of points. - What do you notice about the \(y\)-coordinates of \(K\) and \(M\)? - Can symmetry identify the vertex before you solve for the equation? - Fractional coefficients are valid results.

1. The slopes are \(m_{KL} = \frac{-2 - 1}{1 - (-2)} = -1\) and \(m_{LM} = \frac{1 - (-2)}{4 - 1} = 1\). Since the slopes are different, the points are not collinear. 2. Points \(K\) and \(M\) have the same \(y\)-coordinate and are equally spaced from \(x = 1\), so \(L(1, -2)\) is the vertex. 3. Use vertex form: \(f(x) = a(x - 1)^2 - 2\). 4. Substitute \(K(-2, 1)\): \(1 = a(-3)^2 - 2\), so \(9a = 3\) and \(a = \frac{1}{3}\). 5. Expand: \(f(x) = \frac{1}{3}(x - 1)^2 - 2 = \frac{1}{3}x^2 - \frac{2}{3}x - \frac{5}{3}\).

The points are not collinear. The quadratic function is \(f(x) = \frac{1}{3}x^2 - \frac{2}{3}x - \frac{5}{3}\).

- Substitute the three points into \(g(x) = ax^2 + bx + c\). - The point \((0, 0)\) immediately determines \(c\). - Use \(x = -\frac{b}{2a}\) to find each vertex. - The horizontal range is the positive zero.

1. Let \(g(x) = ax^2 + bx + c\). Since \(P(0, 0)\) lies on the graph, \(c = 0\). 2. Substituting \(Q(5, 4)\) and \(R(10, 6)\) gives \(25a + 5b = 4\) and \(100a + 10b = 6\). Solving yields \(a = -0.04\) and \(b = 1\), so \(g(x) = -0.04x^2 + x\). 3. For Rocket A, the vertex occurs at \(x = -\frac{1}{2(-0.05)} = 10\), and \(f(10) = 5\). Its vertex is \((10, 5)\). 4. For Rocket B, the vertex occurs at \(x = -\frac{1}{2(-0.04)} = 12.5\), and \(g(12.5) = 6.25\). Its vertex is \((12.5, 6.25)\). Rocket B reaches a greater height. 5. Factoring gives \(f(x) = x(-0.05x + 1)\), so its positive zero is \(20\). Also, \(g(x) = x(-0.04x + 1)\), so its positive zero is \(25\). Rocket B travels farther.

a) \(g(x) = -0.04x^2 + x\) b) Rocket A: vertex \((10, 5)\); Rocket B: vertex \((12.5, 6.25)\). Rocket B goes higher. c) Rocket A: \(20\,\text{m}\); Rocket B: \(25\,\text{m}\). Rocket B travels farther.