Exploring the Function f(x) = x² − 6x + 10 Step by Step

A quadratic function follows this general form, where a, b, and c are constants.In our specific example, we have f of x equals x squared minus six x plus ten.The squared term, x squared, is the main component that gives us our parabola shape. Since a is positive one, the parabola opens upward.When we add the linear term, negative six x, it pulls the parabola to the right, influencing where the axis of symmetry will be.Finally, adding the constant term of positive ten shifts the entire parabola up by ten units.Let's summarize how each component affects our quadratic function.The sign of a determines if the parabola opens upward or downward. The b term influences where the axis of symmetry falls, and c shifts the entire graph up or down.Now that we understand the components, we're ready to explore the key points of our quadratic function.For our quadratic function f of x equals x squared minus 6x plus 10, let's find its key points.To find the axis of symmetry, we use the formula x equals negative b over 2a. In our case, b is negative 6 and a is 1.This gives us x equals 3 for our axis of symmetry. Let's visualize this with a vertical line.To find the vertex, we plug x equals 3 back into our original function.When we simplify this calculation...We get y equals 1, making our vertex point three comma one.Next, let's find the y-intercept by plugging in x equals zero.Simplifying this...We get y equals 10, giving us our y-intercept at zero comma ten.These three key elements - the axis of symmetry, vertex, and y-intercept - give us the essential structure of our parabola.Using our key points, we can now visualize the complete parabola.First, let's plot our vertex at (3,1), which we found is the minimum point of our parabola.Next, we have our y-intercept at (0,10), where the parabola crosses the y-axis.Using the quadratic formula, we find our x-intercepts at x equals 2 and x equals 4.Now we can draw our complete parabola, which passes through all these points with perfect symmetry.This quadratic function has many practical applications in the real world.In physics, projectile motion follows a parabolic path, where height changes with time, just like our function.In business, profit functions often form parabolas, helping find the optimal price point for maximum profit.Architects use parabolic shapes in designs, like in suspension bridges, where the cables form natural parabolic curves.