### Determine graph of quadratic equation using various method

Question Sample Titled 'Determine graph of quadratic equation using various method'

If $-{1}<{a}<{0}$, which of the following may represent the graph of ${y}={\left({a}{x}+{1}\right)}^{{2}}+{a}$ ?

A

${x}$${y}$
B

${x}$${y}$
C

${x}$${y}$
D

${x}$${y}$

 ${y}$ $={\left({a}{x}+{1}\right)}^{{2}}+{a}$ ${y}$ $={\left[{a}{\left({x}+\dfrac{{1}}{{a}}\right)}\right]}^{{2}}+{a}$ Factorize out ${a}$ ${y}$ $={a}^{{2}}{\left({x}+\dfrac{{1}}{{a}}\right)}^{{2}}+{a}$ In the form of ${y}={a}{\left({x}-{h}\right)}^{{2}}+{k}$ ∵  ${a}^{{2}}>{0}$ ∴   The graph of the equation opens upwards . ∵   The vertex of the graph of the equation = ${\left(-\dfrac{{1}}{{a}},{a}\right)}$ , where ${a}$ is negative Given $-{1}<{a}<{0}$ ∴   The ${x}$coordinate of the vertex of the graph is positive and ${y}$ coordinate is negative.  ∴   The answer is a graph which opens upwards, ${x}$coordinate of the vertex being positive and ${y}$ coordinate being negative.

 ${y}$ $={\left({a}{x}+{1}\right)}^{{2}}+{a}$ ${y}$ $={a}^{{2}}{x}^{{2}}+{2}{a}{x}+{1}+{a}$ ${y}$ $={a}^{{2}}{x}^{{2}}+{2}{a}{x}+{\left({1}+{a}\right)}$ ∵  ${a}^{{2}}>{0}$ ∴   The graph of the equation opens upwards .  Product of roots $=\dfrac{{{1}+{a}}}{{{a}^{{2}}}}$ Product of roots of equation ${a}{x}^{{2}}+{b}{x}+{c}={0}$ is $\dfrac{{c}}{{a}}$ ∵  ${a}^{{2}}>{0}$ and ${1}+{a}>{0}$ Given $-{1}<{a}<{0}$ ∴   The product of roots is positive.  ∴   The answer is a graph which opens upwards and product of roots being positive.  (Candidates can also determine the answer by observing the ${y}$ intercept and ${x}$ intercept.)

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