Chapter 3: Q. 102 (page 158)

Why does the graph of a quadratic function open up if $a > 0$ and down if $a < 0$ ?

Short Answer

Expert verified

When $a > 0$ then $f (x)$ approaches positive infinity and hence it opens upwards

when $a < 0$ then $f (x)$ approaches negative infinity and hence it opens downwards.

Step by step solution

Given information

Consider a quadratic equation $f (x) = a {(x - h)}^{2} + k$

When a is positive

Now we evaluate,

$f (\infty) = a {(\infty - h)}^{2} + k f (\infty) = a {(\infty)}^{2} + k f (\infty) = \infty A l s o f (- \infty) = a {(- \infty - h)}^{2} + k f (- \infty) = a {(- \infty)}^{2} + k f (- \infty) = \infty$

So when $a > 0$ then $f (x)$ approaches positive infinity

When a is negative

Now we evaluate

$f (\infty) = - a {(\infty - h)}^{2} + k f (\infty) = - a {(\infty)}^{2} + k f (\infty) = - \infty A l s o f (- \infty) = - a {(- \infty - h)}^{2} + k f (- \infty) = - a {(- \infty)}^{2} + k f (- \infty) = - \infty$

When $a < 0$ then $f (x)$ approaches negative infinity

Conclusion

When $a > 0$ then $f (x)$ approaches positive infinity and hence it opens upwards Similarly when $a < 0$ then $f (x)$ approaches negative infinity and hence it opens downwards.

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x	$y = f (x)$
-2	$\frac{1}{4}$
-1	$\frac{1}{2}$
0	1
1	2
2	4

Why does the graph of a quadratic function open up if $a > 0$ and down if $a < 0$ ?

Short Answer

Step by step solution

Given information

When a is positive

When a is negative

Conclusion

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Why does the graph of a quadratic function open up if a>0and down if a<0?

Short Answer

Step by step solution

Given information

When a is positive

When a is negative

Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Theoretical and Mathematical Physics

Decision Maths

Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Why does the graph of a quadratic function open up if $a > 0$ and down if $a < 0$ ?