Question

What is the y intercept of the quadratic function f(x)=x^2+x-6

Answer 1

Answer:

$(0,\, -6)$.

Step-by-step explanation:

On a cartesian plane, the $y$-intercept of a function is the point where the graph of that function intersects with the $y\!$-axis.

The $y$-axis of a cartesian plane is the same as the equation $x = 0$ (that is, the collection of all points with an $x$-coordinate of $0$.)

Construct a system of two equations, with one equation representing $y$-axis and $y = f(x)$ to represent the graph of this function:

$\begin{aligned}\begin{cases} y = x^{2} + x - 6 & \text{for the quadratic function} \\ x = 0 & \text{for the y -axis}\end{cases}\end{aligned}$.

Solve this system for $x$ and for $y$. If a solution exists, then the $y\!$-axis and the graph of $y = f(x)$ would indeed intersect. The point $(x,\, y)$ would be the intersection of the $y\!\!$-axis and the graph of $y = f(x)\!$.

Substitute the second equation of the system into the first.

$\begin{aligned}\begin{cases} x = 0 \\ y = -6\end{cases}\end{aligned}$.

Hence, the intersection of the $y$-axis and the graph of $y = f(x)$ would be $(0,\, -6)$. By definition, this point would be the $y\!$-intercept of $y = f(x)\!$.