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Consider the graphs of a quadratic function and a linear function on the same set of axes. cutting2 points of intersection $b^2-4ac \gt 0$ touching1 point of intersection $b^2-4ac = 0$ missingno points of intersection $b^2-4ac \lt 0$ In the graphs meet, the coordinates of the points of intersection of the graphs can be found […]
Example 1 Find the equation of the quadratic graph. \( \begin{align} \displaystyley &= a(x+2)(x+1) \\2 &= a(0+2)(0+1) &\text{substitute }(0,2)\\2 &= 2a \\a &= 1 \\y &= 1(x+2)(x+1) \\\therefore y &= x^2+3x+2 \end{align} \) Example 2 Find the equation of the quadratic graph. \( \begin{align} \displaystyley &= a(x+1)^2 \\1 &= a(0+1)^2 &\text{substitute }(0,1)\\a &= 1 \\y […]
The equation of the axis of symmetry of $y=ax^2+bx+c$ is $x=-\dfrac{b}{2a}$. Example 1 Find the equation of the axis of symmetry of $y=x^2+4x-2$. \( \begin{align} \displaystylex &= -\dfrac{b}{2a} \\&= -\dfrac{4}{2 \times 1} \\\therefore x &= -2\end{align} \) Example 2 Find the equation of the axis of symmetry of $y=-2x^2-12x+3$. \( \begin{align} \displaystylex &= -\dfrac{b}{2a} \\&= […]
$$ \large y=a(x-b)(x-c)$$Concave up for $a \gt 0$ Concave down for $a \lt 0$ Example 1 Draw the graph of $y=(x-1)(x+2)$. \( \begin{align} \displaystyle(x-1)(x+2) & = 0 \\x-1 &= 0 \text{ or } x+2 = 0 \\\therefore x &= 1 \text{ or } x = -2 \end{align} \) Example 2 Draw the graph of $y=(x+1)(x-2)$. […]
$$ \large y=(x-a)^2+b$$ Example 1 Draw the graph of $y=(x-1)^2+2$. The vertex is $(1,2)$, and the graph is concave up. Example 2 Draw the graph of $y=(x-1)^2-2$. The vertex is $(1,-2)$, and the graph is concave up. Example 3 Draw the graph of $y=(x+1)^2+2$. The vertex is $(-1,2)$, and the graph is concave up. Example […]
The process of finding the maximum or minimum value of functions is called optimisation.For the quadratic function $y=ax^2+bx+c$, we have already seen that the vertex has $x$-coordinate $-\dfrac{b}{2a}$. $a>0$: the minimum value of $y$ occurs at $x=-\dfrac{b}{2a}$ $a<0$: the maximum value of $y$ occurs at $x=-\dfrac{b}{2a}$ We need to identify a situation’s maximum or minimum […]
The Discriminant and the Quadratic Graph The discriminant of the quadratic equation $ax^2+bx+c=0$ is $\Delta = b^2-4ac$. We used the discriminant to determine the number of real roots of the quadratic equation. If they exist, these roots correspond to $x$-intercepts of the quadratic $y = ax^2+bx+c$. The discriminant tells us about the relationship between a […]
There are many situations in science, engineering, economics and other fields where quadratic equations are a vital part of the mathematics used. Some real word problems can be solved using a quadratic equation. We are generally only interested in any real solutions which result. We will look at some of these applications. We employ the […]
In many cases, factorising a quadratic equation or completing the square can be long or difficult. We can instead use the quadratic formula. \( \begin{align} \displaystyle \require{AMSsymbols} \require{color}ax^2 + bx + c &= 0 \\ax^2 + bx &= -c \\x^2 + \dfrac{b}{a}x &= -\dfrac{c}{a} \\x^2 + \dfrac{b}{a}x \color{red} + \Big(\dfrac{b}{2a}\Big)^2 &= -\dfrac{c}{a} \color{red} + \Big(\dfrac{b}{2a}\Big)^2\\\Big(x+\dfrac{b}{2a}\Big)^2 […]
Not all quadratics factorise easily. For instance, $x^2+6x+2$ cannot be factorised by simple factorisation. In other words, we cannot write $x^2+6x+2$ in the form $(x-a)(x-b)$ where $a$ and $b$ are rational. An alternative way to solve this equation $x^2+6x+2$ is by completing the square. Equations of the form $ax^2+bx+c$ can be converted to the form […]