A parabola associated with a certain quadratic function f...

A parabola associated with a certain quadratic function $f$ has the point (2,8) as its vertex and passes through the point $(4,0) .$ Find an expression for $f(x)$ in the form $f(x)=a(x-h)^{2}+k$ (a) From the given information, find the values of $h$ and $k$ (b) Substitute the values you found for $h$ and $k$ into the expression $f(x)=a(x-h)^{2}+k$ (c) Now find $a$. To do this, use the fact that the parabola passes through the point $(4,0) .$ That is, $f(4)=0$ You should get an equation having just $a$ as a variable. Solve for $a$ (d) Substitute the value you found for $a$ into the expression you found in part (b). (e) Graph the function using a graphing utility and check your answer. Is (2,8) the vertex of the parabola? Does the parabola pass through (4,0)$?$

A parabola associated with a certain quadratic function $f$ has the point (2,8) as its vertex and passes through the point $(4,0) .$ Find an expression for $f(x)$ in the form $f(x)=a(x-h)^{2}+k$
(a) From the given information, find the values of $h$ and $k$
(b) Substitute the values you found for $h$ and $k$ into the expression $f(x)=a(x-h)^{2}+k$
(c) Now find $a$. To do this, use the fact that the parabola passes through the point $(4,0) .$ That is, $f(4)=0$ You should get an equation having just $a$ as a variable. Solve for $a$
(d) Substitute the value you found for $a$ into the expression you found in part (b).
(e) Graph the function using a graphing utility and check your answer. Is (2,8) the vertex of the parabola? Does the parabola pass through (4,0)$?$

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Key Concepts

Determining Parameters Using Substitution

When additional points on the parabola are provided, substituting these coordinates into the vertex form f(x) = a(x - h)² + k allows one to solve for the coefficient a. This process ensures that the quadratic function accurately passes through the given points.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is written as f(x) = a(x - h)² + k, which clearly shows the vertex (h, k) of the parabola. This form is particularly useful when the vertex is known, as it directly incorporates the coordinates of the vertex, simplifying further analysis and transformations.

Vertex of a Parabola

The vertex of a parabola is the point at which the function achieves its maximum or minimum value. In the vertex form, (h, k) represents this point, making it easier to determine the axis of symmetry of the parabola and to understand how the graph is shifted from the origin.

Graphical Verification

Graphical verification involves plotting the quadratic function to ensure that the calculated vertex and passing points are correct. This step helps validate the algebraic solution by providing a visual confirmation that the parabola has the correct shape and position.

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