Question

SKETCHING GRAPHS Sketch the graph of the function. Label the vertex. $$ y=-2 x^{2}+6 x-5 $$

Short answer

Begin with a sketch that represents the general form of a parabola. Next, place the vertex at the point (1.5, -0.5), and sketch the graph passing through the y-intercept point (0, -5). Since the function does not have real x-intercepts, it does not cross the x-axis.

Step-by-step solution

Identify the Function Form

The given function is a quadratic function, which is typically in the form \(y = ax^{2} + bx + c\). Comparing this general equation with our function \(y = -2x^{2} + 6x - 5\), we can see that \(a = -2\), \(b = 6\) and \(c = -5\). Since \(a\) is negative, the graph opens downwards.

Locate the Vertex

The vertex of a quadratic function \(y = ax^{2} + bx + c\) can be found using the formula \(h = -b / 2a\), where \(h\) represents the x-coordinate of the vertex. Substituting \(a = -2\) and \(b = 6\) into the formula, we get \(h = -6 / 2(-2) = 1.5\). To find the y-coordinate of the vertex, we substitute \(h = 1.5\) into our function, obtaining \(y = -2(1.5)^{2} + 6(1.5) - 5 = -0.5\). Therefore, the vertex of the function is at (1.5, -0.5).

Plot the Y-intercept and Vertex

The vertex together with y-intercept provide two points that can set the direction for the curve of the graph. The y-intercept is the point where the parabola crosses the y-axis, and it is simply the constant term in our given function, which is \(c = -5\). Therefore, the y-intercept is at (0, -5). Now plot the points (1.5, -0.5) representing the vertex and (0, -5) for y-intercept on the graph.

Find X-intercepts

The x-intercepts are the points where the graph crosses the x-axis i.e., when \(y = 0\). To find these points we need to solve the equation \(-2x^{2} + 6x - 5 = 0\). This can be done using the quadratic formula \(x = \[-b ± \sqrt{b^{2} - 4ac} / 2a\]\). However, in our case, the discriminant (\(b^{2} - 4ac\)) is negative (which is less than zero), indicating no real solutions. Hence, there are no x-intercepts.

Sketch the Function

Now use all the information obtained to sketch the graph of the function. Draw a downward-opening parabola passing through the points corresponding to the vertex and the y-intercept.

Key concepts

Quadratic Function

A quadratic function is typically expressed in the standard form \( y = ax^2 + bx + c \) where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. This form is key to characterizing the shape and direction of a parabola, which is the graph of a quadratic function. If \( a \) is positive, the parabola opens upwards; if \( a \) is negative, as in our exercise with \( a = -2 \), the parabola opens downwards.

The graph's width and direction are determined by this coefficient; the larger the absolute value of \( a \), the narrower the parabola. Understanding this concept is essential for sketching the graph correctly.
Moreover, the \( b \) value influences the horizontal placement of the vertex, while \( c \) tells us the y-intercept, which is the point where the graph crosses the y-axis.

Vertex of a Parabola

The vertex is a crucial concept when dealing with parabolas. It is the highest or lowest point on the graph of a quadratic function, depending on whether the parabola opens upwards or downwards. In the solution, we calculated the vertex of the parabola with the given function \( y = -2x^2 + 6x - 5 \) using the formula \( h = -\frac{b}{2a} \) for the x-coordinate and then substituting \( h \) back into the original equation to find the y-coordinate.

To find the \( y \)-value of the vertex, it's vital to substitute the calculated \( h \) back into the original quadratic equation. For our problem, the vertex is located at (1.5, -0.5), providing a reference point for sketching the parabola on a graph.

Y-Intercept

The y-intercept is the point where the parabola crosses the y-axis. It is another essential feature when sketching quadratic graphs. This intercept can be quickly identified as it corresponds to the value of \( c \) in the quadratic function \( y = ax^2 + bx + c \).

In the given equation, the y-intercept is the constant term \( c = -5 \), hence the parabola crosses the y-axis at the point (0, -5). When you plot this point, along with the vertex, it begins to give shape to the parabola and can assist in ensuring accuracy when drawing the curve.

X-Intercepts

The x-intercepts are where the graph intersects with the x-axis. These are found by setting \( y \) to zero and solving the quadratic equation. For many quadratics, this solves to two points, where the parabola hits the x-axis.

However, not all quadratic functions have real x-intercept points. The discriminant of the quadratic formula, \( b^2 - 4ac \), determines whether the solutions are real and how many there are. In our exercise, the discriminant was negative, indicating no real x-intercepts are present. In such cases, the parabola does not touch the x-axis at all, and the curve is either completely above or below it, depending on the direction of opening.