4J - Rules of quadratic graphs
Families of quadratic graphs:
A family of quadratic functions will all share a common feature, or common features, while other parameters are used to indicate that many different equations exist within the family. Examples include:
A family of quadratic functions will all share a common feature, or common features, while other parameters are used to indicate that many different equations exist within the family. Examples include:
- The equation \(y=ax^2+bx+4\) describes a family of parabolas that all have a \(y\)-intercept at \((0,4)\).
- The equation \(y=a(x+3)(x-b)\) describes a family of parabolas that all have an \(x\)-intercept at \((-3,0)\).
- The equation \(y=a(x+1)^2+4\) describes a family of parabolas that all have a turning point at \((-1,4)\).
- The equation \(y=ax^2+bx\) describes a family of parabolas that all have at least one \(x\)-intercept at \((0,0)\).
4J - Example 1: Families of quadratic graphs
A family of quadratics all have a vertex existing on the line \(x=3\), the general equation for this family is \(y=a(x-3)^2+k\). Determine the equation of the member which has a range of \([5,\infty)\) and passes through the point \((5,11)\).
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4J - Example 1: Video solution
4J - Example 1: Practice
Question 1: ABC Question 2: ABC 4J - Example 1: Solutions
Question 1: ABC Question 2: ABC |
4J - Example 2: Families of quadratic graphs
A family of quadratics with two \(x\)-intercepts, one being the origin, has the general equation \(y=ax^2+bx\).
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4J - Example 2: Video solution
4J - Example 2: Practice
Question 1: ABC Question 2: ABC 4J - Example 2: Solutions
Question 1: ABC Question 2: ABC |
Determining the rule of a quadratic:
When determining the rule of a quadratic you need to be able to identify what information you are given about the function/graph. Possible information includes:
When determining the rule of a quadratic you need to be able to identify what information you are given about the function/graph. Possible information includes:
- The \(y\)-intercept
- The \(x\)-intercept(s)
- The turning point
- Point(s) on the curve.
- The general form: \(y=ax^2+bx+c\)
- Factorised form: \(y=a(x-m)(x-n)\)
- Turning point form: \(y=a(x-h)^2+k\)
Case 1: If you are given any three points on the parabola
If you are given any three points on a parabola, use the general quadratic form \(y=ax^2+bx+c\). To find the equation:
If you are given two \(x\)-intercepts, \((m,0)\) and \((n,0)\), and another point \((x_1,y_1)\), use the factorised form \(y=a(x-m)(x-n)\). To find the equation:
If you are given the turning point, \((h,k)\), and another point, \((x_1,y_1)\), use the turning point form \(y=a(x-h)^2+k\). To find the equation:
If you are given any three points on a parabola, use the general quadratic form \(y=ax^2+bx+c\). To find the equation:
- Substitute the 3 points, one at a time, into the \(y=ax^2+bx+c\) to obtain 3 equations with 3 unknowns (\(a\), \(b\), and \(c\)).
- Find the values of \(a\), \(b\) and \(c\) by solving the system of equations simultaneously.
- State the full equation by substituting the values of \(a\), \(b\) and \(c\) into \(y=ax^2+bx+c\).
If you are given two \(x\)-intercepts, \((m,0)\) and \((n,0)\), and another point \((x_1,y_1)\), use the factorised form \(y=a(x-m)(x-n)\). To find the equation:
- Substitute the value of the \(x\)-intercepts into the factorised form. Remember that the signs will be opposite in the equation due to the null factor law.
- Substitute the other point, \((x_1,y_1)\), into the equation and solve for the parameter \(a\).
- State the full equation by substituting the values of \(a\), \(m\) and \(n\) into \(y=a(x-m)(x-n)\).
If you are given the turning point, \((h,k)\), and another point, \((x_1,y_1)\), use the turning point form \(y=a(x-h)^2+k\). To find the equation:
- Substitute the value of the turning point, \((h,k)\), into the turning point form.
- Substitute the other point, \((x_1,y_1)\), into the equation and solve for the parameter \(a\).
- State the full equation by substituting the values of \(a\), \(h\) and \(k\) into \(y=a(x-h)^2+k\).
■ Any three points will define a unique parabola; however, if you are given the turning point, only one additional point is required to fully define the parabola.
4J - Example 3: Determining the rule of a parabola given three points
Determine the rule for the quadratic which goes through the points \((0,1)\), \((1,1)\) and \((2,2)\).
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4J - Example 3: Video solution
4J - Example 3: Practice
Question 1: ABC Question 2: ABC 4J - Example 3: Solutions
Question 1: ABC Question 2: ABC |
4J - Example 4: Determining the rule of a parabola given the turning point
Determine the rule for the quadratic with a turning point at \((1,2)\) and passing through another point \((-1,9)\).
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4J - Example 4: Video solution
4J - Example 4: Practice
Question 1: ABC Question 2: ABC 4J - Example 4: Solutions
Question 1: ABC Question 2: ABC |
4J - Example 5: Determining the rule of a parabola given the x-intercepts
Determine the rule for the quadratic with \(x\)-intercepts at \(x=1\) and \(x=2\) and passes through another point \((0,3)\).
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4J - Example 5: Video solution
4J - Example 5: Practice
Question 1: ABC Question 2: ABC 4J - Example 5: Solutions
Question 1: ABC Question 2: ABC |
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Rules of quadratic graphs - Worksheet A
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