Solve the equation $x^2 - x - 6 = 0.$ The graphs of four quadratic functions are shown below - Leaving Cert Mathematics - Question (a)(b)(c) - 2014

The correct graph for the function $f(x) = x^2 - x - 6$ can be identified by finding the vertex and the intercepts. The vertex lies at $x = \frac{-b}{2a} = \frac{1}{2}$. The function opens upwards, and the curve intersects the x-axis at $x = -2$ and $x = 3$. Upon examining the provided graphs,

Graph D accurately depicts these features.

To sketch $h(x)$, we take the graph of $g(x)$ and shift it up by 2 units. Therefore, the entire graph of $g(x)$ will be displaced vertically upwards. This includes moving the vertex from (1, 1) to (1, 3). The new points after the upward shift will represent the graph of $h(x)$.

To sketch $k(x)$, we need to shift the graph of $g(x)$ horizontally to the left by 2 units. This means if the original vertex of $g(x)$ is at (1, 1), the vertex of $k(x)$ will now be at (-1, 1). Plot the new vertex and ensure that the shape of the parabola remains consistent while adjusting the x-coordinates accordingly.