Compare $g(x)$ to the general vertex form $y = a(x-h)^2 + k$. In that form, what does $(h, k)$ represent on the graph?

Look at the sign of the coefficient in front of $(x-1)^2$. If that coefficient is negative, does the parabola open upward or downward, and does that give you a maximum or a minimum at the vertex?

In Desmos, type g(x) = -2(x-1)^2 + 7 to graph the function.

On the graph, locate the highest point of the parabola. You can click or tap on that point, or use the "maximum" feature in Desmos (for example, by typing maximum(g(x), x1, x2) with an appropriate interval).

From the vertex that Desmos shows, note the y-coordinate. That y-value is the maximum value of $g(x)$.

The function is given by $g(x) = -2(x-1)^2 + 7$. This is a quadratic function written in vertex form:

In this form, the graph is a parabola with vertex at $(h, k)$ and it opens upward if $a>0$ and downward if $a<0$.

In $g(x) = -2(x-1)^2 + 7$, the coefficient $a$ is $-2$, which is less than 0. That means the parabola opens downward, so it has a maximum (not a minimum) at its vertex.

So the maximum value of $g(x)$ is the y-coordinate of the vertex of this parabola.

Match $g(x) = -2(x-1)^2 + 7$ to the general form $y = a(x-h)^2 + k$:

• $a = -2$
• $h = 1$ (because of $(x-1)$)
• $k = 7$

So the vertex is at $(1, 7)$, and because the parabola opens downward, the maximum value of $g(x)$ is the y-coordinate of the vertex.

Therefore, the maximum value of $g(x)$ is 7.