Question

what is the slope of y=x^2+x+7​

Answer 1

A quadratic function doesn't have a slope, because the function is not constantly increasing.

The slope exists if the graph is a straight line, so functions with slopes are called a linear function.

The slope is not a single value (It has a variable)

But the idea of linear functions is used in calculus. The slope is a ratio of how much value increased and x.

If the value of x increased by a very little value Δx, we can write the slope.

(Increased graph value in the numerator and x value in the denominator.)

Let's use $y=x^2+x+7$  for example.

$f(x+\Delta x)=(x+\Delta x)^2+(x+\Delta x)+7$ is the greater value.

Let's expand it.

$f(x+\Delta x)=x^2+2x\Delta x+(\Delta x)^2+x+\Delta x+7$

$f(x)=x^2+x+7$

So, the increased value will be the difference.

The difference is $2x\Delta x+(\Delta x)^2+\Delta x$.

The slope will be $\dfrac{2x\Delta x+(\Delta x)^2+\Delta x}{\Delta x} =2x+\Delta x+1$.

The value of $\Delta x$ is so little that we can ignore it.

Hence slope will be $2x+1$. (It is different from the linear equation because slope has a variable.)

Tangents of quadratic functions?

If we use the slope, at x=2 the slope will be 5. Since the tangent passes (2, 13), the tangent will be $y=5x+3$.

I hope this was enough. If you want to graph any function try desmos.