Question

Sketch the graph of the function. Label the vertex. $$ y=2 x^{2}-8 x+3 $$

Short answer

The vertex of the function \(y = 2x^{2} - 8x + 3\) is at the point (2, -5). The graph of the function is a upward facing parabola.

Step-by-step solution

Find the Vertex

The vertex of a quadratic function is given as \(\left(h, k\right)\) where \(h\) is the x-coordinate and \(k\) is the y-coordinate. The x-coordinate \(h\) is given as \(h = -\frac{b}{2a}\). For our given equation \(y = 2x^2 - 8x + 3\), \(a = 2\) and \(b=-8\). Plugging these values in, we get \( h = -\frac{-8}{2(2)} = 2 \). To find \(k\), replace \(x\) with \(h\) in the original equation which gives: \(k = 2(2)^2 -8(2) + 3 = -5\). Therefore, the vertex is \((2, -5)\).

Sketch the Graph

Sketch a graph with the vertex at the point (2, -5). Remember, this function is a upward facing parabola because the coefficient of \(x^2\) is positive. It reaches its minimum value at the vertex. Plot a few more points for accuracy and draw the parabola.

Key concepts

Vertex of a Parabola

The vertex of a quadratic function's graph, known as a parabola, is a critical concept for students to grasp. It's the point at which the parabola either reaches its maximum or minimum value. When graphing quadratic functions, identifying the vertex is the first step to plotting the curve accurately.

To find the vertex, you can use the formula \( h = -\frac{b}{2a} \) for the x-coordinate, where \( a \) and \( b \) are coefficients from the quadratic equation in its standard form \( y = ax^2 + bx + c \). You then evaluate the original function at \( x = h \) to find the y-coordinate, \( k \) of the vertex. Following this process allows students to locate the vertex which serves as a pivotal reference point in sketching the parabola.

Quadratic Equation

Understanding the quadratic equation \( y = ax^2 + bx + c \) is foundational for graphing parabolas. This equation represents a second-degree polynomial, where \( a, b, \) and \( c \) are constants and \( x \) and \( y \) are variables. The coefficient \( a \) determines whether the parabola opens upwards (if \( a > 0 \) or downwards (if \( a < 0 \)) and affects the width of the parabola. The coefficient \( b \) influences the horizontal placement, and \( c \) represents the y-intercept where the graph crosses the y-axis. Familiarity with this equation enables students to manipulate and graph quadratic functions effectively.

Parabola Sketching

Sketching a parabola involves several steps, starting with identifying the vertex, as previously discussed. Once the vertex is located, you can plot the y-intercept found by setting \( x=0 \) in the equation. Students should note that the parabola is symmetrical around its axis of symmetry. This means that points on opposite sides of the axis are equidistant from it. After plotting the vertex and y-intercept, sketch additional points on both sides of the axis of symmetry for accuracy. By connecting these points, a smooth U-shaped curve (for \( a > 0 \) or an inverted U (for \( a < 0 \) can be drawn to represent the parabola.

Axis of Symmetry

The axis of symmetry of a parabola is a vertical line that passes through the vertex, effectively dividing the parabola into two mirror-image halves. For a quadratic equation in the form \( y = ax^2 + bx + c \) the axis of symmetry's equation is \( x = h \) where \( h \) is the x-coordinate of the vertex. Knowing the axis of symmetry is crucial for graphing because it ensures that all plotted points will reflect the parabola's symmetrical nature. The recognition of this important feature guides students in understanding the underlying beauty and consistency in the structure of quadratic functions.