Question

Graph the function. Label the \(x\)-intercept(s) and the \(y\)-intercept. \(h(x)=-x^2+5 x-6\)

Short answer

The \(x\)-intercepts are \(x=2\) and \(x=3\), and the \(y\)-intercept is \(y=-6\). The function can now be graphed with these intercepts.

Step-by-step solution

Find the \(x\)-intercepts

To find the \(x\)-intercepts, set \(h(x)\) to zero and solve for \(x\). It gives: \[0=-x^2+5x-6\]. This can be factored as \(0=-1 * (x-2)(x-3)\]. Setting each factor equal to zero gives the two \(x\)-intercepts: \(x=2\) and \(x=3\).

Find the \(y\)-intercept

To find the \(y\)-intercept, set \(x=0\) in the function, so \(h(0)=-0^2+5*0-6=-6\), which gives the \(y\)-intercept: \(y=-6\)

Graph the function

After finding the \(x\) and \(y\)-intercepts, the graph of the function can be sketched. The parabola will open downwards since the coefficient of \(x^2\) is negative. Place the \(x\)-intercepts at \(x=2\) and \(x=3\), and the \(y\)-intercept at \(y=-6\).

Key concepts

X-Intercepts

When graphing quadratic functions, identifying the points where the graph intersects the x-axis is a crucial step. These intersection points are known as the x-intercepts, or the solutions to the equation when the function's value is zero. To find the x-intercepts for the quadratic function h(x) = -x^2 + 5x - 6, we set the function equal to zero and solve for x.

In step 1 of the solution, by factoring the quadratic, we determine the intercepts to be x = 2 and x = 3. Knowing the x-intercepts assists in sketching the overall shape of the graph and is foundational in understanding the function's behavior.

Y-Intercept

In contrast to the x-intercepts, the y-intercept of a quadratic function is the point where the graph crosses the y-axis. This occurs when x is zero. It's a single point and helps anchor the graph on the coordinate plane. Calculating the y-intercept is straightforward: simply substitute zero for x in the function.

For the equation h(x) = -x^2 + 5x - 6, substituting zero, we get h(0) = -6, revealing the y-intercept to be (0, -6). This, alongside the x-intercepts, provides another critical reference for accurately plotting the parabola.

Factoring Quadratics

The step of factoring quadratics is fundamental in finding the roots or zero-points of a quadratic equation. Factoring is the process of breaking down the quadratic equation into a product of simpler binomials or integers. It allows us to solve for the x-intercepts easily.

For the function given, h(x) = -x^2 + 5x - 6, factoring yields (x - 2)(x - 3) after taking out a common factor of -1. The equation is then set to zero, and the 'Zero Product Property' leads us to the x-intercepts. Factoring is a powerful technique, especially when quadratics can be easily broken down into their binomial factors.

Parabola

A parabola is the characteristic 'U' shaped curve formed by graphing a quadratic function. The direction in which the parabola opens is determined by the leading coefficient of the quadratic term. If the coefficient is positive, the parabola opens upwards, and if negative, as in the function h(x) = -x^2 + 5x - 6, it opens downwards.

The shape of the parabola is symmetric about a vertical line called the axis of symmetry. For the given function, since it opens downwards, the vertex is at the topmost point of the curve, and the x-intercepts mark where the curve hits the x-axis. Each of these key parts of the parabola—the vertex, axis of symmetry, direction, and intercepts—offer information about the function's graph and serve as a guide for drawing it accurately.