Quadratic Functions: Miscellaneous Exercise (Grade 10, New Syllabus)

ဖုန်းဖြင့်ကြည့်သည့်အခါ စာများကို အပြည့်မမြင်ရလျှင် screen ကို ဘယ်ညာ ဆွဲကြည့်နိုင်ပါသည်။

Important Notes

Quadratic Function (Standard Form)	$f(x)=a x^{2}+b x+c, a \neq 0$
Graph	Parabola $a>0$ (opens upward) $a<0$ (opens downward)
Axis of Symmetry	$x=-\displaystyle\frac{b}{2 a}$
Vertex	$\left(-\displaystyle\frac{b}{2 a}, f\left(-\displaystyle\frac{b}{2 a}\right)\right)=\left(-\displaystyle\frac{b}{2 a},-\displaystyle\frac{b^{2}-4 a c}{4 a}\right)$
y-intercept	(0, c)
Discriminant	$b^{2}-4 a c$ $b^{2}-4 a c>0 \Rightarrow$ two $x$ intercepts (cuts $x-$ axis at two points) $b^{2}-4 a c=0 \Rightarrow$ one $x$ intercepts (touch $x$ -axis at one point $)$ $b^{2}-4 a c=0 \Rightarrow$ one $x$ intercepts (does not intersect $x$ -axis)
Quadratic Equation	$a x^{2}+b x+c=0, a \neq 0$
Quadratic Formula	$x=\displaystyle\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$
Quadratic Function (Vertex Form)	$f(x)=a(x-h)^{2}+k, a \neq 0$
Vertex	$(h, k)$
Axis of Symmetry (Vertex Form)	$x=h$
Quadratic Function (Intercept Form when discriminant $>0$)	$f(x)=a(x-p)(x-q), a \neq 0$
$X$ -intercept points	$(p, 0)$ and $(q, 0)$
Axis of Symmetry	$x=\displaystyle\frac{p+q}{2}$
Quadratic Inequality	$a x^{2}+b x+c>0$ $a x^{2}+b x+c \geq 0$ $a x^{2}+b x+c<0$ $a x^{2}+b x+c \leq 0$
$a>0$ and $b^2-4ac<0$	The graph does not cut the $x$ -axis. $y<0 \Rightarrow$ solution set $=\varnothing$ $y=0 \Rightarrow$ solution set $=\varnothing$ $y>0 \Rightarrow$ solution set $=\mathbb{R}$
$a<0$ and $b^2-4ac<0$	The graph does not cut the $x$ -axis. $y<0 \Rightarrow$ solution set $=\mathbb{R}$ $y=0 \Rightarrow$ solution set $=\varnothing$ $y>0 \Rightarrow$ solution set $=\varnothing$
$a>0$ and $b^2-4ac=0$	The graph touches the $x$ -axis. $y<0 \Rightarrow$ solution set $=\varnothing$ $y=0 \Rightarrow$ solution set $=\left\{-\displaystyle\frac{b}{2 a}\right\}$ $y>0 \Rightarrow$ solution set $=\mathbb{R} \backslash\left\{-\displaystyle\frac{b}{2 a}\right\}$
$a<0$ and $b^2-4ac=0$	The graph touches the $x$ -axis. $y<0 \Rightarrow$ solution set $=\mathbb{R} \backslash\left\{-\displaystyle\frac{b}{2 a}\right\}$ $y=0 \Rightarrow$ solution set $=\left\{-\displaystyle\frac{b}{2 a}\right\}$ $y>0 \Rightarrow$ solution set $=\varnothing$
$a>0$ and $b^2-4ac>0$	The graph cuts the $x$ -axis at two points. $y<0 \Rightarrow$ solution set $=\{x \mid p<x<q\}$ $y=0 \Rightarrow$ solution set $=\{p, q\}$ $y>0 \Rightarrow$ solution set $=\{x \mid x<p$ or $x>q\}$
$a<0$ and $b^2-4ac>0$	The graph cuts the $x$ -axis at two points. $y<0 \Rightarrow$ solution set $=\{x \mid x<p$ or $x>q\}$ $y=0 \Rightarrow$ solution set $=\{p, q\}$ $y>0 \Rightarrow$ solution set $=\{x \mid p<x<q\}$

Exercises

ယခု post တွင်ပါဝင်သော မေးခွန်းများသည် grade 10 သင်ရိုးသစ်ပြဌာန်းချက်နှင့် သက်ဆိုင်သော သင်ရိုးလေ့ကျင့်ခန်း ပြင်ပမေးခွန်းများ ဖြစ်ပြီး Target Mathematics Grade 10 (Volume 1) တွင် Miscellaneous Exercise အဖြစ်ထည့်သွင်းပေးခဲ့ပါသည်။

1. The figure shows the graph of $y=x^2$ shifted to four new positions. Write an equation for each new graph.

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2. The figure shows the graph of $y=-x^2$ shifted to four new positions. Write an equation for each new graph.

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3. Find the standard form of the equation for the quadratic function for each diagram shown below.

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4. A quadratic function has an equation in the form $y=x^2+ax+a$ and passes through the point (1, 9). Calculate the value of $a$.

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5. The graph of quadratic function $y=ax^2+bx+c$ passes through the points (1,1), (0, 0) and (−1,1). Calculate the value of $a, b$ and $c$.

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6. A parabola has its vertex at the point $V (1,1)$ and passes through the point $(0,2)$. Find its equation in the form of $y=x^2+bx+c$.

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7. The graph of the function $f(x)=ax^2+bx+c$ has vertex at $(1, 4)$ and passes through the point $(−1,−8)$. Find $a, b$ and $c$.

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8. Given that $y = x^2 + 2x − 8$.
   

   (a)	What is the vertex of $y$?
   (b)	What are the $x$-intercepts of the graph of $y$ ?
   (c)	Find the $x$-coordinates of the ponts on the graph $y = x^2 + 2x − 8$ when $y = −8$.


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9. If a piece of string of fixed length is made to enclose a rectangle, show that the enclosed area is the greatest when the rectangle is a square.

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10. A farmer has $2000$ yards of fence to enclose a rectangular field. What are the dimensions of the rectangle that encloses the largest area?

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11. The profit (in millions of kyats) of a company is given by $P(x)= 5000+1000x−5x^2$ where $x$ is the amount ( in millions of kyats) the company spends on advertising.
    

    (a)	Find the amount, $x$, that the company has to spend to maximize its profit.
    (b)	Find the maximum profit.


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12. Find the integral values of $x$ that satisfy the inequality $x^2 + 48 < 16x$.

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13. Find the suitable domain of the function $f(x) = 1 + 3x − 2x^2$ for which the curve of $f(x)$ lies completely above the line $y = −1$.
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