Consider the following quadratic functions. Without graphing...

Consider the following quadratic functions. Without graphing them, answer the questions below. a) $f(x)=2 x^{2}$ b) $f(x)=-x^{2}$ c) $f(x)=\frac{1}{4} x^{2}$ d) $f(x)=-5 x^{2}+3$ e) $f(x)=\frac{2}{3}(x-1)^{2}-3$ f) $f(x)=-2(x+3)^{2}+1$ g) $f(x)=(x-3)^{2}+1$ h) $f(x)=-4(x+1)^{2}-3$ Which graph has vertex $(-3,1) ?$

Consider the following quadratic functions. Without graphing them, answer the questions below.
a) $f(x)=2 x^{2}$
b) $f(x)=-x^{2}$
c) $f(x)=\frac{1}{4} x^{2}$
d) $f(x)=-5 x^{2}+3$
e) $f(x)=\frac{2}{3}(x-1)^{2}-3$
f) $f(x)=-2(x+3)^{2}+1$
g) $f(x)=(x-3)^{2}+1$
h) $f(x)=-4(x+1)^{2}-3$
Which graph has vertex $(-3,1) ?$

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two that can be expressed in a standard form ax² + bx + c. They represent parabolas when graphed and are characterized by a symmetric curve with a vertex that serves as the maximum or minimum point, depending on the direction in which the parabola opens.

Vertex Form

The vertex form of a quadratic function, given as f(x) = a(x - h)² + k, directly reveals the vertex of the parabola at the point (h, k). This form simplifies the process of identifying key features of the graph, such as the vertex, axis of symmetry, and the effect of transformations applied to the function.

Graph Transformations

Graph transformations involve altering the graph of a parent function through operations such as translations, reflections, stretching, or compressing. In the context of quadratics, the vertex form makes it easy to see how the graph is shifted horizontally and vertically from the origin, as well as how the coefficient 'a' affects the width and orientation of the parabola.

Identifying the Vertex

Identifying the vertex is crucial in analyzing quadratic functions, as it indicates the highest or lowest point on the graph, depending on whether the function opens downwards or upwards. When a quadratic function is in vertex form, the coordinates of the vertex can be directly read off, allowing for immediate analysis of the function's graph without the need for additional operations such as completing the square.

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