Question
Consider the Quadratic function
.
Its vertex is
Its largest
-intercept is
.
Its
-intercept is
Its vertex is
Its largest
Its
Ask by Guzman Chang. in the United States
Mar 16,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The vertex is
, the largest
-intercept is
, and the
-intercept is
.
Solution
We start with the quadratic function
1. Finding the vertex
The vertex
of a quadratic function
is given by
Here,
and
, so
Next, we find
by evaluating
:
Calculate
:
Now, compute the rest:
Thus,
So, the vertex is
2. Finding the
-intercepts
To find the
-intercepts, set
:
This quadratic factors as
Setting each factor equal to zero gives:
which implies
The largest
-intercept is
3. Finding the
-intercept
The
-intercept is found by evaluating
:
Thus, the
-intercept is
Summary
- The vertex is
. - The largest
-intercept is . - The
-intercept is .
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The Deep Dive
The vertex of the quadratic function
can be found using the vertex formula
. Here,
and
, giving us
. To find the corresponding
-coordinate, substitute
back into the function. The vertex is at
.
Next, to find the largest
-intercept, we can use the quadratic formula given by
. Plugging in values here gives us two
-intercepts:
and
. The largest is
.
Lastly, to find the
-intercept, we evaluate the function at
:
. So the
-intercept is
.
In summary:
- Vertex:
- Largest
-intercept: -
-intercept: