Without using your calculator, match the given equations...

Without using your calculator, match the given equations with the appropriate graph. State which equation does not match any of the graphs. Explain.
$$\begin{array}{l}y=x^{2}-6 x+10 \\y=x^{2}-3 x-4 \\y=-0.3 x^{2}+2 x+3 \\y=-3 x^{2}+12 x-8\end{array}$$

Key Concepts

Discriminant and Roots

The discriminant, given by b² - 4ac, provides information about the nature and number of the roots of a quadratic equation. It indicates whether the parabola intersects the x-axis at two points, touches it at one point, or does not cross it at all. Recognizing these features helps in matching the correct equation to a graph based on the x-intercepts and overall graph behavior.

Graph Transformations

Transformations involve shifting, reflecting, and stretching or compressing the graph of a basic quadratic function. By completing the square to rewrite the quadratic in vertex form, one can easily identify these transformations. This is useful for matching equations to graphs by recognizing translations (movements along the axes) and changes in the curvature.

Parabola Concavity and Width

The sign and magnitude of the coefficient a in the quadratic function not only determine whether the parabola opens upward or downward, but also affect its 'width' or steepness. A larger absolute value of a results in a narrower parabola, while a smaller absolute value produces a wider one. This concept is key when comparing the given equations to different graphical representations.

Vertex and Axis of Symmetry

The vertex of a quadratic function is the maximum or minimum point of the parabola, found by using the formula x = -b/(2a). The axis of symmetry is a vertical line passing through the vertex, and it divides the parabola into two mirror-image halves. Determining the vertex helps in accurately sketching and matching the graph of a quadratic function.

Quadratic Functions

A quadratic function is a second-degree polynomial expressed in the form y = ax² + bx + c, where the coefficient a determines the direction (upward if positive, downward if negative) and the 'width' or steepness of the parabola. Understanding this form is essential for predicting the general shape and properties of the resulting graph.

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