Problem 29 - Introductory Algebra for College Students

Question

Graph the parabola whose equation is given $$y=-x^{2}+4 x-3$$

Step-by-Step Solution

Answer

The graph of the equation \(y = -x^2 + 4x - 3\) is a parabola with vertex at \((2, -1)\), which opens downwards and crosses the y-axis at \((0, -3)\).

Step 1: Determine the Vertex

The vertex of a parabola given in the standard form \(y = ax^2 + bx + c\) is located at \((-\frac{b}{2a} , f(-\frac{b}{2a}))\). Applying this for the equation \(y=-x^2+4x-3\), we get \(\left(\frac{4}{2*(-1)}, -\left(\frac{4}{2*(-1)}\right) ^2+4\left(\frac{4}{2*(-1)}\right)-3\right)\), which simplifies to \((2, -1)\). So the vertex of the parabola is at \((2, -1)\).

Step 2: Direction of Opening

The direction in which the parabola opens depends on the sign of the leading coefficient \(a\). If \(a\) is positive, the parabola opens upwards, and if \(a\) is negative, it opens downwards. For the equation \(y=-x^2+4x-3\), the leading coefficient is -1, thus the parabola will open downwards.

Step 3: Find the Y-Intercept

The y-intercept is the point where the parabola crosses the y-axis. This occurs when \(x = 0\). For our equation we get \(y=-0^2+4*0-3\) which simplifies to \(-3\). Thus the equation intersects the y-axis at the point \((0,-3)\).

Step 4: Plot the Graph

With the vertex \((2, -1)\), the knowledge that the parabola opens downwards, and the y-intercept \((0, -3)\), we can plot the parabola. First, draw a dot for the vertex and the y-intercept. Next, because of the symmetry of a parabola, plot a point that is directly to the right or left of the vertex and follows the opening direction. Now connect these points making sure that the graph is wider at the top or bottom.

Key Concepts

Vertex of a Parabola

Understanding the vertex of a parabola is crucial for graphing it correctly. The vertex is essentially the 'tip' of the parabola, the point where it changes direction. For a parabola in the standard form of \(y = ax^2 + bx + c\), you can locate the vertex by using the formula for its coordinates \(\left(-\frac{b}{2a} , f(-\frac{b}{2a})\right)\). Let's break this down with our example equation, \(y=-x^2+4x-3\). We identify \(a=-1\), \(b=4\), and apply these values to find the x-coordinate of the vertex: \(-\frac{b}{2a} = -\frac{4}{2(-1)} = 2\). Next, we substitute this x-coordinate back into the original equation to find the corresponding y-coordinate, ending up with the vertex \( (2, -1) \).

The location of the vertex not only helps in graphing the parabola accurately but also gives us vital information about its maximum or minimum value depending on the direction in which it opens. Remember, the vertex is the key feature that shapes the parabola's curve.

Direction of Parabola Opening

The direction in which a parabola opens is determined by the coefficient \(a\) of the \(x^2\) term in the quadratic equation. When this leading coefficient is positive, as in \(y = ax^2\), the parabola opens upwards, much like a bowl resting right-side up. Conversely, if the coefficient is negative, the parabola will open downwards, similar to an upside-down bowl.

In our example \(y=-x^2+4x-3\), the leading coefficient is \(a=-1\). This negative sign tells us that the parabola opens downwards. This information is vital for sketching the overall shape of the graph. It also influences the nature of the vertex - since the parabola in the example opens downwards, the vertex represents the highest point on the graph, or the maximum value of the parabola.

Y-Intercept of a Parabola

The y-intercept of a parabola is the point where the curve crosses the y-axis. To find this, we simply look for when \(x=0\) in the equation of the parabola. In practical terms, the y-intercept is where the graph 'starts' on your y-axis when you’re plotting it by hand, or the starting point of the parabola on a graphing tool.

For the equation given in our exercise, \(y=-x^2+4x-3\), when we plug in \(x=0\), we find the y-value to be \(y=-0^2+4*0-3 = -3\). Thus, the parabola intersects the y-axis at the point \( (0,-3) \). This point is a crucial part of plotting the graph accurately, as it will be one of the first points you place onto the coordinate plane before drawing the curve. Moreover, it can serve as a check for accuracy once the rest of the parabola is drawn.