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Graphing rational functions can be a fun challenge because these graphs often have unique shapes due to their asymptotes and intercepts. A rational function is a fraction where both the numerator and the denominator are polynomials. In our case, the function is given by \[ y = \frac{4x^2 - 100}{2x^2 + x - 15} \] To start graphing this rational function, you want to determine several key features: x-intercepts, y-intercepts, and asymptotes. These will help guide the shape of the graph. The graph will approach certain lines (asymptotes), cross the axes at intercepts, and have its own distinctive look in different sections of the coordinate plane.

Asymptotes are lines that the graph of the function gets closer to but never actually touches. There are two main types: vertical and horizontal.- **Vertical Asymptotes:** Occur where the denominator of the rational function is zero and the function itself is undefined. To find this, solve the equation \[ 2x^2 + x - 15 = 0.\] These values of \(x\) make the function approach infinity.- **Horizontal Asymptotes:** Found by comparing the leading terms of the numerator and the denominator, especially when \(x\) becomes very large or very small. For the function \[ y = \frac{4x^2 - 100}{2x^2 + x - 15}, \] the leading term in the numerator is \(4x^2\) and in the denominator is \(2x^2\). Thus, the horizontal asymptote will be \[ y = \frac{4}{2} = 2. \] This indicates that as \(x\) grows large, the graph levels out near \(y = 2\). Remember that horizontal asymptotes are approached as \(x\) heads toward positive or negative infinity.

X-intercepts of a function are the points where the graph crosses the x-axis. To find x-intercepts, set the numerator equal to zero. In our function, set \[ 4x^2 - 100 = 0. \] The solutions to this equation are the x-coordinates of the intercepts.Step by step, solve the equation like this:- First, add 100 to both sides:\[ 4x^2 = 100.\]- Then, divide both sides by 4:\[ x^2 = 25.\]- Now, take the square root of both sides:\[ x = \pm 5. \] These solutions mean there are x-intercepts at \((5, 0)\) and \((-5, 0)\). The graph will cross the x-axis at these points.

Y-intercepts are where the graph crosses the y-axis. Unlike x-intercepts, there is only one y-intercept for a function. To find it, set \(x = 0\) in the equation of the function. For \[ y = \frac{4x^2 - 100}{2x^2 + x - 15}, \] substitute zero for \(x\) in both the numerator and the denominator:- Numerator becomes \[ 4(0)^2 - 100 = -100.\]- Denominator becomes \[ 2(0)^2 + 0 - 15 = -15.\] Thus, \[ y = \frac{-100}{-15} = \frac{20}{3}. \] So, the y-intercept is at \[ \left( 0, \frac{20}{3} \right). \] The graph will cross the y-axis at this point, giving us another key marker for constructing the graph.