Question

graph the equation y = x² - 10x + 24 on the accompanying set of axes. you must plot 5 points including the roots and the vertex. using the graph, determine the roots of the equation x² - 10x + 24 = 0. click to plot points. click points to delete them.

Explanation:

Step1: Find the roots

Set $y = 0$, so $x^{2}-10x + 24=0$. Factor the quadratic equation: $(x - 4)(x - 6)=0$. Then $x-4 = 0$ or $x - 6=0$, giving $x=4$ and $x = 6$.

Step2: Find the vertex

For a quadratic function $y=ax^{2}+bx + c$ (here $a = 1$, $b=-10$, $c = 24$), the $x$-coordinate of the vertex is $x=-\frac{b}{2a}=-\frac{-10}{2\times1}=5$. Substitute $x = 5$ into the equation $y=x^{2}-10x + 24$, we get $y=5^{2}-10\times5 + 24=25-50 + 24=-1$. So the vertex is $(5,-1)$.

Step3: Choose additional points

Let $x=3$, then $y=3^{2}-10\times3 + 24=9-30 + 24=3$. Let $x=7$, then $y=7^{2}-10\times7 + 24=49-70 + 24=3$. Let $x=2$, then $y=2^{2}-10\times2 + 24=4-20 + 24=8$.

Answer:

Roots are $x = 4$ and $x = 6$, vertex is $(5,-1)$. Additional points: $(3,3)$, $(7,3)$, $(2,8)$