Question

A standard parabola in the \(x y\)-coordinate plane intersects the \(x\)-axis at \((5,0)\) and \((-5,0)\). What is the value of the \(x\)-coordinate of this parabola's line of symmetry?

Short answer

The value of the x-coordinate of the parabola's line of symmetry is 0.

Step-by-step solution

Identify the x-intercepts#

The given parabola intersects the x-axis at (5, 0) and (-5, 0). These two points are the x-intercepts of the parabola.

Find the midpoint of the x-intercepts#

The line of symmetry's x-coordinate is found at the midpoint between the x-intercepts. We can use the midpoint formula: \[x_m = \frac{x_1 + x_2}{2}\] Where \(x_m\) is the x-coordinate of the midpoint, \(x_1\) is the x-coordinate of the first point, and \(x_2\) is the x-coordinate of the second point. Plugging in the given values, we get: \[x_m = \frac{5 + (-5)}{2}\]

Simplify and find the x-coordinate of the line of symmetry#

Calculating the value of the x-coordinate of the line of symmetry using the values found in step 2: \[x_m = \frac{5 + (-5)}{2} = \frac{0}{2} = 0\] The value of the x-coordinate of the parabola's line of symmetry is 0.

Key concepts

X-Intercepts of a Parabola

Understanding the x-intercepts of a parabola is crucial in the realm of algebra and SAT problem-solving. An x-intercept is a point where a graph crosses or touches the x-axis. In simpler terms, it's where the output value, or the y-coordinate, is zero. For a standard parabola, which is symmetrical and U-shaped, there could be up to two x-intercepts, one, or none at all, depending on how the parabola sits in relation to the x-axis.

With the given exercise example, the parabola's x-intercepts are at the coordinates \(5, 0\) and \( -5, 0\). The symmetry of a parabola indicates that these intercepts are equidistant from the parabola's axis of symmetry. This axis is a vertical line that splits the parabola into two mirror images. The x-axis coordinates of the intercepts help us ascertain this exact line, as it lies precisely in the middle of them.

Finding the x-intercepts can also assist in graphing the parabola and determining its vertex, which is the highest or lowest point on the graph.

Midpoint Formula

The midpoint formula is a fundamental tool in coordinate geometry, which allows us to find the exact center point between two given points. It effectively gives us the average of the x-coordinates and y-coordinates of those points to yield the coordinates of the midpoint. The standard midpoint formula is \[ x_{m} = \frac{x_{1} + x_{2}}{2} \] and \[ y_{m} = \frac{y_{1} + y_{2}}{2} \] where \(x_{m}, y_{m}\) represent the midpoint's coordinates, and \(x_{1}, y_{1}\) and \(x_{2}, y_{2}\) are the coordinates of the original points.

In the context of our exercise, we only need to focus on the x-coordinates since the y-coordinate for both intercepts is zero, which, after applying the midpoint, confirms that the x-coordinate of the midpoint, and hence the line of symmetry, is zero. This formula isn't just handy for geometry; it's also widely applicable across various mathematical concepts, including statistics and even computer graphics.

SAT Math Problems

The SAT math section is designed to assess a student's ability to reason and solve problems using mathematical concepts. SAT math problems typically cover a range of topics from algebra, geometry, trigonometry, to data analysis. It's important for students to be familiar with concepts such as finding x-intercepts and calculating midpoints, as these can often form the basis of SAT questions.

In practice, SAT math problems require a good mix of conceptual understanding and procedural skill. For instance, being given the x-intercepts of a parabola on the SAT, you might be asked to determine the parabola's equation, axis of symmetry, or vertex. Developing strategies such as quickly sketching a graph or breaking down the problem into more manageable steps are extremely helpful in tackling these questions.

The exercise at hand is typical of the SAT math section, both in terms of content and the logical steps required to reach the solution. By mastering the key topics involved, students can feel more confident and prepared for these standardized tests.