Question
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Answer
The quadratic function
Solution
Step 1. Write Down the Given Equation
The quadratic function is given by
Step 2. Complete the Square
To express the quadratic in vertex form, we complete the square.
-
Factor out the coefficient of
from the first two terms: -
Complete the square for the expression in the parentheses. We have:
-
Substitute back into the equation:
-
Distribute the
: -
Combine the constant terms:
Step 3. Identify the Vertex
The vertex form of a quadratic function is
where the vertex is
Thus, the vertex is
Step 4. Determine the Axis of Symmetry
The axis of symmetry for a quadratic equation in vertex form is given by
So, the axis of symmetry is
Step 5. Find the Zeros (x-intercepts)
To find the zeros, set
Solve for
-
Add
to both sides: -
Divide both sides by
: -
Take the square root of both sides:
-
Solve for
:
Thus, the zeros are
Summary
-
The vertex form of the quadratic is
-
The vertex is
. -
The axis of symmetry is
. -
The zeros (x-intercepts) are
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Simplify this solution Bonus Knowledge
Let’s dive into the world of quadratic functions! The equation you provided,
Now, to see this equation in action, consider its applications in areas such as physics, where quadratic functions can describe the motion of objects under the influence of gravity. For example, if you throw a ball into the air, its trajectory can be modeled by a quadratic equation like yours. By analyzing the function, you can determine the ball’s maximum height and how long it’ll take to hit the ground!
