Graphing Quadratics: Plotting f(x)=x²+10x+24

Let's examine our quadratic function f of x equals x squared plus ten x plus twenty-four.This function is written in standard form, which follows the pattern a x squared plus b x plus c.Let's identify each component. The coefficient of x squared, a, equals 1. The coefficient of x, b, equals 10. And our constant term, c, equals 24.These coefficients are crucial as they determine the shape and position of our parabola.Since a is positive - specifically, a equals 1 - we know our parabola will open upward, like a cup or a smile.To graph this parabola accurately, we'll need to find two key features: the axis of symmetry and the vertex.Each term in our quadratic function plays a specific role. The x squared term determines the opening direction and width, the x term causes horizontal shifting, and the constant term moves the parabola up or down.Now that we understand the components of our quadratic function, we're ready to find its key points.To graph our quadratic function accurately, we need to find several key points.Let's start by finding the vertex. We'll use the formula x equals negative b over two a.Substituting our values, where b is 10 and a is 1...Simplifying...Now we'll find the y-coordinate by plugging x equals negative 5 back into our original function.Simplifying the calculation...This gives us our vertex point at negative 5, negative 1.To find the x-intercepts, we need to factor the quadratic equation and set it equal to zero.The expression factors to x plus 4 times x plus 6 equals zero.Solving for x, we get x equals negative 4 or x equals negative 6.These give us our x-intercepts at negative 6, zero and negative 4, zero.Finally, let's find the y-intercept by evaluating f of zero.Substituting zero for x...We get y equals 24.This gives us our y-intercept at zero, twenty-four.We now have all the key points needed to graph our parabola accurately.Now that we have our key points calculated, let's plot them on the coordinate plane.First, let's plot our vertex at negative five comma negative one.Next, we'll add our x-intercepts at negative six comma zero and negative four comma zero.Our y-intercept is at zero comma twenty-four.The axis of symmetry passes through our vertex at x equals negative five.Now we can draw our parabola, connecting all these points with a smooth curve.Notice how the curve is perfectly symmetrical around the axis. Any point on one side has a matching point on the other side.Let's review the key features of our graph. The parabola opens upward because a is positive, the vertex is our lowest point, and we have perfect symmetry around x equals negative five.And there we have it - a complete and accurate graph of our quadratic function f of x equals x squared plus ten x plus twenty-four.Thanks for graphing this quadratic function with Spark.E!