Graph each function. See Examples 6–8. h(x) = 2x² - 1

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Recognize that the given function $h(x) = 2x^2 - 1$ is a quadratic function, which graphs as a parabola.

Identify the key features of the parabola: the coefficient of $x^2$ is positive (2), so the parabola opens upwards.

Find the vertex of the parabola. Since the function is in the form $h(x) = ax^2 + bx + c$ with $a=2$, $b=0$, and $c=-1$, the vertex's x-coordinate is given by $x = -\frac{b}{2a} = 0$.

Calculate the y-coordinate of the vertex by substituting $x=0$ into the function: $h(0) = 2(0)^2 - 1 = -1$. So, the vertex is at $(0, -1)$.

Plot the vertex and additional points by choosing values of $x$ (for example, $x=1$ and $x=-1$), calculate $h(x)$ for these values, and then sketch the parabola opening upwards through these points.

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Understanding Quadratic Functions

A quadratic function is a polynomial of degree two, generally expressed as f(x) = ax² + bx + c. Its graph is a parabola, which opens upward if a > 0 and downward if a < 0. Recognizing the shape and key features of quadratics is essential for graphing.

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Vertex and Axis of Symmetry

The vertex of a parabola is its highest or lowest point, found using the formula x = -b/(2a). The axis of symmetry is a vertical line through the vertex that divides the parabola into two mirror images. Identifying these helps in accurately sketching the graph.

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To graph a quadratic function, calculate and plot key points such as the vertex, y-intercept (where x=0), and x-intercepts (roots). These points guide the shape and position of the parabola on the coordinate plane.

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