Plug $x=-1$ and $x=2$ into $y=2^x+x$ to get the corresponding $y$-values. Be careful with the negative exponent in $2^{-1}$ and with adding the $x$ term.

Once you have the two points, use $m = \dfrac{y_2-y_1}{x_2-x_1}$ to find the slope. Simplify the fraction carefully, especially when combining fractions in the numerator.

In one expression line, type 2^(-1) + (-1) to get the $y$-value at $x=-1$. In another expression line, type 2^2 + 2 to get the $y$-value at $x=2$. Note both results.

In a new expression line, type (6 - (-1/2)) / (2 - (-1)) but replace 6 and -1/2 with the exact numerical results you saw in step 1, if they differ. The value that Desmos outputs for this expression is the slope of line $t$.

The line $t$ intersects the graph of $y=f(x)=2^x+x$ at $x=-1$ and $x=2$. That means line $t$ passes through the two points whose coordinates are $(x, f(x))$ at those $x$-values. The slope of line $t$ is the slope of the line through these two points.

Compute $f(-1)$ and $f(2)$.

• For $x=-1$:

  • $2^{-1} = \dfrac{1}{2}$
  • $f(-1) = 2^{-1} + (-1) = \dfrac{1}{2} - 1 = -\dfrac{1}{2}$ So one point is $(-1, -\tfrac{1}{2})$.

• For $x=2$:

  • $2^{2} = 4$
  • $f(2) = 4 + 2 = 6$ So the other point is $(2, 6)$.

The slope $m$ of a line through points $(x_1, y_1)$ and $(x_2, y_2)$ is

Using $(x_1, y_1)=(-1, -\tfrac{1}{2})$ and $(x_2, y_2)=(2, 6)$, we get

Simplify the numerator and denominator separately:

• Numerator:

• Denominator:

So the slope is

Thus, the slope of line $t$ is $\dfrac{13}{6}$.