Question

The quadratic function $f(x)=a(x-h)^2+k$ is in standard form. (a) The graph of $f$ is a parabola with vertex (___,_) (b) If $a>0$, the graph of $f$ opens _ . In this case $f(h)=k$ is the value _ of $f$. (c) If $a<0$, the graph of $f$ opens _ . In this case $f(h)=k$ is the value ___ of $f$.

Name: Problem 2, Chapter 3: Precalculus: Mathematics for Calculus
Uploaded: 2019-06-29T04:27:42-08:00
Duration: 1 min 43 s
Channel: Vishnu Rathnam
Description: The quadratic function f(x)=a(x-h)^2+k is in standard form. (a) The graph of f is a parabola with vertex (,) (b) If a>0, the graph of f opens . In this case f(h)=k is the value of f. (c) If a<0, the graph of f opens . In this case f(h)=k is the value of f.

   The quadratic function $f(x)=a(x-h)^2+k$ is in standard form.
(a) The graph of $f$ is a parabola with vertex (_____,_____)
(b) If $a>0$, the graph of $f$ opens _____ . In this case $f(h)=k$ is the value _____ of $f$.
(c) If $a<0$, the graph of $f$ opens _____ . In this case $f(h)=k$ is the value _____ of $f$.

Precalculus: Mathematics for Calculus

James Stewart,… 6th Edition

Chapter 3, Problem 2

Instant Answer

Solved by Expert Vishnu Rathnam

06/29/2019

Step 1

So, in this case, the vertex of the parabola represented by the function $f(x)=a(x-h)^2+k$ is $(h,k)$. Show more…

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The quadratic function $f(x)=a(x-h)^2+k$ is in standard form. (a) The graph of $f$ is a parabola with vertex (_____,_____) (b) If $a>0$, the graph of $f$ opens _____ . In this case $f(h)=k$ is the value _____ of $f$. (c) If $a<0$, the graph of $f$ opens _____ . In this case $f(h)=k$ is the value _____ of $f$.

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Key Concepts

Quadratic Function (Vertex Form)

This is a way to represent a quadratic function that makes it easy to identify the vertex of the parabola directly. The standard vertex form is expressed as f(x) = a(x - h)² + k, where (h, k) represents the vertex.

Vertex

The vertex is the point where the parabola changes direction, serving as the highest or lowest point on the graph depending on the orientation of the parabola. In the vertex form, this point is given by (h, k).

Coefficient a and Direction of Opening

The value of the coefficient a determines the width of the parabola and its direction. If a is greater than zero, the parabola opens upward, and if a is less than zero, it opens downward.

Extremum (Minimum/Maximum Value)

The vertex of the parabola also represents the extremum of the quadratic function. Specifically, if a > 0, the vertex is the point of minimum value, and if a < 0, the vertex is the point of maximum value.

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