Question

SKETCHING GRAPHS Sketch the graph of the function. Label the vertex. $$ y=-4 x^{2}+4 x+7 $$

Short answer

The vertex of the function \(y=-4 x^{2}+4 x+7\) is at \((1,7)\), and the axis of symmetry is the line \(x = 1\). The graph of the function is a downward parabola that opens down with vertex at \((1,7)\) and passes through the y-axis at \(7\).

Step-by-step solution

Identify the Vertex

The vertex of a quadratic equation in standard form is given by the formula \((-b/2a, f(-b/2a))\), where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation, and \(f(x)\) is the function of \(x\). Here, \(a = -4\), \(b = 4\), and \(c = 7\). Therefore, the x-coordinate of the vertex is \(-b/2a = -4/2(-4) = 1\). Substituting \(x = 1\) into the function gives the y-coordinate: \(y = -4*(1)^2 + 4*1 + 7 = -4 + 4 + 7 = 7\). So, the vertex is \((1,7)\).

Identify the Axis of Symmetry

The axis of symmetry is the vertical line that passes through the vertex. Its equation is given by \(x = -b/2a\), which we calculated in the previous step as \(x = 1\). Therefore, the equation for the axis of symmetry is \(x = 1\).

Sketch the Graph

First, plot the vertex at \((1,7)\) and draw the axis of symmetry at \(x = 1\). Then, plot any other two symmetric points in the graph, for example, the y-intercept at \(x = 0\) and at \(x = 2\). Then, connect the points using a curve. This will give the sketch of the function.

Key concepts

Vertex of a Quadratic Equation

The vertex of a quadratic equation represents the highest or lowest point on its graph, which is called a parabola. Finding this point is crucial when sketching quadratic graphs because it serves as a cornerstone for the parabola's shape and direction.

For the quadratic equation in standard form, which is \(y = ax^2 + bx + c\), the vertex \( (h, k) \) can be found using the formula \(h = -\frac{b}{2a}\) and \(k = f(h)\). Here, \(a\), \(b\), and \(c\) are the coefficients that determine the parabola's properties, with \(a\) influencing the direction of opening, \(b\) influencing the vertex's position, and \(c\) representing the y-intercept.

In our example, \( a = -4 \) and \( b = 4 \) imply that the parabola opens downwards (as \( a < 0 \) ) and the vertex lies along the equation \( x = 1 \). The coordinate \( 1 \) for \( x \) helps us calculate the \( y \) value, which is \(7\). Thus, the vertex is \( (1,7) \), indicating the parabola's peak and helping us visualize the graph's extent and curvature.

Axis of Symmetry

The axis of symmetry is an imaginary line that divides the parabola into two mirroring halves. It's a vital concept in sketching quadratic graphs because every parabola is symmetrical about this axis.

For any quadratic equation \( y = ax^2 + bx + c \), the axis of symmetry's equation is \( x = -\frac{b}{2a} \). This axis runs vertically through the vertex of the parabola, which means every point on one side of the axis has a corresponding point with equal distance on the other side.

In our exercise, since \( b = 4 \) and \( a = -4 \) , the axis of symmetry is the line \( x = 1 \). It tells us that for every point \( (x, y) \) on the parabola, there will be another point \( (2-x, y) \) that is directly across from it on the other side of this line. By using the axis of symmetry, we ensure that our graph of the parabola is accurate and balanced, a key aspect of properly sketching the complete figure.

Graphing Parabolas

Graphing parabolas involves plotting a U-shaped curve where every point satisfies the quadratic equation \( y = ax^2 + bx + c \). Understanding how to graph these curves properly will allow students not just to complete assignments, but also to analyze and predict the behavior of quadratic functions.

To sketch the parabola accurately, begin by plotting the vertex - which in our given problem is \( (1,7) \). Next, draw the axis of symmetry, the line \( x = 1 \) in our case. After establishing these guideposts, determine a few points on either side of the parabola by substituting \( x \) values into the equation to find corresponding \( y \) values, and plot these points on the graph. Due to the symmetry, you only need to calculate points on one side of the axis and reflect them across to the other side.

Remember, the distance from the vertex to these points, and vice versa across the axis of symmetry, should be equal. Once enough points are plotted, draw a smooth curve connecting them, ensuring that it reflects the correct direction that the parabola opens (upwards if \( a > 0 \), downwards if \( a < 0 \)). This guided approach to graphing will make it easier for the students to grasp the visual aspect of quadratic equations and understand their graphical representations.