Determine whether the statement is true or false. Justify your answer. The functions f(x)=3 x^2+

step 1 Alright everyone in this question we have to tell whether given statement is true or false. So f

Determine whether the statement is true or false. Justify...

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Key Concepts

Quadratic Functions

A quadratic function is a second-degree polynomial typically expressed in the form f(x) = ax² + bx + c. Its graph is a parabola, and the value of the coefficient 'a' determines whether it opens upward or downward. Understanding the structure of quadratic functions is key to analyzing many properties of their graphs, including symmetry, intercepts, and the vertex.

Vertex of a Parabola

The vertex of a parabola is the point where the curve attains its maximum or minimum value. It can be calculated using the formula x = -b/(2a) for the x-coordinate, and then finding the corresponding y-coordinate by substituting this x-value back into the function. The vertex is also the point of symmetry for the graph, meaning the shape of the parabola is mirrored on either side of this point.

Effect of the Constant Term

In the quadratic function f(x) = ax² + bx + c, the constant term 'c' causes a vertical shift of the graph. Changing 'c' moves the parabola up or down without affecting the x-coordinate of the vertex. When two quadratic functions share the same quadratic and linear coefficients, their vertices will have the same x-coordinate, and the vertical shift only changes the y-coordinate by a constant amount.

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