The given quadratic expression is:

$y = x^2 + 8x + 7$

Behavior of the function as $x$ increases and decreases:

1. As $x$ increases (positive values of $x$):

   • The term $x^2$ becomes increasingly large.
   • The linear term $8x$ also increases but at a slower rate compared to $x^2$.
   • As a result, the $y$-values increase rapidly as $x$ increases.

2. As $x$ decreases (negative values of $x$):

   • The quadratic term $x^2$ still grows, but it grows positively since squaring any number gives a positive result.
   • The term $8x$ becomes more negative (since $x$ is negative), but the $x^2$ term dominates for large negative values.
   • Therefore, the $y$-values still increase as $x$ decreases significantly in magnitude.

Conclusion:

• As $x$ increases or decreases significantly, $y$-values grow larger due to the dominance of the $x^2$ term. The graph of this quadratic function is a parabola opening upwards, meaning it has a minimum point (vertex) and then rises as you move in either direction away from the vertex.

Do you want further details or explanations about this?

Here are 5 related questions:

1. What is the vertex of the quadratic function?
2. How do you find the axis of symmetry for a parabola?
3. How does the sign of the coefficient of $x^2$ affect the direction of the parabola?
4. Can you find the roots of the quadratic equation?
5. What is the minimum value of this quadratic function?

Tip: The turning point of a parabola (vertex) gives useful information about where the function reaches its minimum or maximum value.