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| Quadratic Functions and Equations |
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Definition |
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A quadratic expression in one variable is an algebraic expression that has the highest power variable is two. |
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- The basic form of a quadratic expression is \(ax^2 + bx + c\), which is \(a, b\, \text{and} \,c\) is a constant and \(a ≠ 0\), \(x\) is a variable.
- \(a\) is the coefficient of \(x^2\), \(b\) is the coefficient of \(x\) and \(c\) is a constant
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Examples |
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\(x^2+5x-1\) |
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\(-y^2+3y\) |
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\(2m^2+7\) |
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Tips |
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Besides \(x\), other letters can be used to represents variables |
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Relationship between a quadratic function and many-to-one relation
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Quadratic function, \(f(x)= ax^2+bx+c \) |
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- All quadratic functions have the same image for two different images
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Shapes of the graph, \(f(x)= ax^2+bx+c , a \neq0\) |
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- For the graph \(a<0\), \((x_1,y_1)\) is known as maximum point
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- For the graph \(a>0\), \((x_2, y_2)\) is known as minimum point
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Tips |
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The curved shape of the graph of a quadratic function is called a parabola |
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| Axis of symmetry of the graph of a quadratic function |
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Explanation |
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Definition: A straight line that is parallel to the \(y-\)axis and divides the graph into two parts of the same size and shape |
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- The axis of symmetry will pass through the maximum and minimum point of the graph of the function as shown in the diagram below
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Equation axis of symmetry , \(x= - \dfrac{b}{2a}\)
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| Effects of changing the values of \(a,b, \) and \(c\) on graphs of quadratic functions, \(f(x)= ax^2 +bx +c\) |
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- The value of \(a\) determines the shape of the graph
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- The value of \(b\) determines the position of the axis of symmetry
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- The value of \(c\) determines the position of the \(y-\)intercept
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| Forming a quadratic equation based on a situation |
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- A quadratic function is written in the form of \(f(x)= ax^2 +bx +c \) while a quadratic equation is written in the general form \(ax^2 +bx +c = 0\)
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| Roots of a quadratic equation |
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- The root of a quadratic equation \(ax^2 +bx +c = 0\) are the values of the variables, \(x\) which satisfy the equation.
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Relationship between the roots of a quadratic equation and the positions of the roots.
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| The roots of equation \(ax^2 +bx +c = 0\) are the points of intersection of the graph of the quadratic functions \(f(x)= ax^2 +bx +c \) and the \(x-\)axis which are also knowns as the \(x-\)intercepts. |
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| Determine the roots of a quadratic equation by using factorisation method |
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Tips |
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A quadratic equation needs to be written in the form of \(ax^2 +bx +c = 0\) before we carry out factorisation |
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Example |
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Determine the roots of this quadratic equations by using factorisation method \(x^2 - 5x + 6 = 0\). |
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Solution:
\(\begin{aligned} &\space x^2 - 5x +6 = 0 \\\\& (x-3)(x-2) = 0 \\\\& x =3 \space \text{or} \space x = 2. \end{aligned}\)
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| Determine the roots of a quadratic equation by using graphical method |
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- The roots of a quadratic equation \(ax^2 +bx +c = 0\) can be obtained by using a graphical method by reading the values of \(x\) which are the points of intersections of the graph
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