| ဖုန်းဖြင့်ကြည့်သည့်အခါ စာများကို အပြည့်မမြင်ရလျှင် screen ကို ဘယ်ညာ ဆွဲကြည့်နိုင်ပါသည်။ |
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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။
- The figure shows the graph of $y=x^2$ shifted to four new positions. Write an equation for each new graph.
- The figure shows the graph of $y=-x^2$ shifted to four new positions. Write an equation for each new graph.
- Find the standard form of the equation for the quadratic function for each diagram shown below.
- 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 $a$.
- 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 $c$.
- 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 $y=x^2+bx+c$.
- 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 $c$.
- 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 $y = −8$. - 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.
- A farmer has $2000$ yards of fence to enclose a rectangular field. What are the dimensions of the rectangle that encloses the largest area?
- 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. - Find the integral values of $x$ that satisfy the inequality $x^2 + 48 < 16x$.
- 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 $y = −1$. [Show Solution ]
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စာဖတ်သူ၏ အမြင်ကို လေးစားစွာစောင့်မျှော်လျက်!