Question

Write a quadratic function for the following characteristics

Answer 1

`y=x^2+6x-16`

Explanation:

`\boxed{\begin{minipage}{6 cm}\underline{Intercept form of a quadratic equation}\\\\$y=a(x-p)(x-q)$\\\\where:\\ \phantom{ww}$\bullet$ $p$ and $q$ are the $x$-intercepts. \\ \phantom{ww}$\bullet$ $a$ is some constant.\\\end{minipage}}`

Given points on the curve:

• (-8, 0)
• (-6, -16)
• (2, 0)

The x-intercepts of a quadratic function are the points at which the line crosses the x-axis, so when y = 0.

Therefore, the x-intercepts of the given function are -8 and 2.

Substitute these into the intercept formula:

`\implies y=a(x-(-8))(x-2)`

`\implies y=a(x+8)(x-2)`

Substitute the other given point (-6, -16) into the equation and solve for a:

`\begin{aligned} y&=a(x+8)(x-2)\\\textsf{When }(-6,-16)\implies -16&=a(-6+8)(-6-2)\\-16&=a(2)(-8)\\-16&=-16a\\\implies a&=1\end{aligned}`

Therefore, the equation of the function in intercept form is:

`y=(x+8)(x-2)`

Expand to standard form:

`\implies y=x(x-2)+8(x-2)`

`\implies y=x^2-2x+8x-16`

`\implies y=x^2+6x-16`

Therefore, the quadratic function in standard form is:

`\boxed{ y=x^2+6x-16}`