Problem 64 - Algebra 1: Concepts and Skills

Question

Sketch the graph of the function. $$y=5 x^{2}+4 x-5$$

Step-by-Step Solution

Answer

The graph of the function \(y = 5x^2 + 4x - 5\) is an upward-opening parabola with y-intercept at -5, vertex at (-0.4, -5.2), and roots at x = -1 and x = 1.

Step 1: Identify the y-intercept

The y-intercept is determined by setting x to 0 in the function. So, \(y = 5(0)^2 + 4(0) - 5\), hence, the y-intercept is -5.

Step 2: Find the Vertex

The vertex is given by the point \((-b/(2a), f(-b/(2a))\), where a and b are the coefficients of \(x^2\) and x respectively, and \(f(x)\) is our function. For this function, a = 5 and b = 4, so the vertex is \((-4/(2*5), f(-4/(2*5))\). Substituting and solving gives us the vertex as (-0.4, -5.2).

Step 3: Find the roots of the function

The roots are found by setting y to 0 and solving for x. Solving the equation \(0 = 5x^2 + 4x - 5\) gives us the roots. Using the quadratic formula \(x = [-b ± sqrt(b^2 - 4ac)] / 2a\), we find the roots to be x = -1 and x = 1.

Step 4: Sketch the graph

From the values of the vertex, the y-intercept, and the roots, we can now sketch the graph of the function. We place the vertex at (-0.4, -5.2), the y-intercept at (0,-5), and mark the roots at x = -1 and x = 1. The graph is a parabola opening upwards.