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Finding the Axis of Symmetry and Vertex of Quadratic Functions

This guide explains how to find the axis of symmetry and vertex of quadratic functions. Using examples, we explore the standard form (y = ax^2 + bx + c), where the parabola's shape (upward or downward) is determined by the sign of (a). The x-coordinate of the vertex is found using the formula (x = -frac{b}{2a}). After finding the vertex, we substitute it back into the function to find the corresponding y-coordinate. Create a graphical representation by plotting points and reflecting them across the axis of symmetry.

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Finding the Axis of Symmetry and Vertex of Quadratic Functions

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  1. a. Find the axis of symmetry of the graph of the function. Find the vertex of the graph of the function. b. EXAMPLE 1 Find the axis of symmetry and the vertex Consider the function y= – 2x2 + 12x – 7.

  2. a. For the function y = –2x2 + 12x – 7, a = 2 and b = 12. x=– 12 b b = =3 2(– 2) 2a 2a b. The x-coordinate of the vertex is , or 3. – ANSWER The vertex is (3, 11). EXAMPLE 1 Find the axis of symmetry and the vertex SOLUTION Substitute – 2 for aand 12 for b. Then simplify. y = –2(3)2 + 12(3) – 7 = 11 Substitute 3 for x. Then simplify.

  3. = = 1. 2(3) b –6 – x = – Find and draw the axis of symmetry: 2a EXAMPLE 2 Graphy = ax2 + bx + c Graphy = 3x2 – 6x + 2. STEP1 Determine whether the parabola opens up or down. Because a > 0, the parabola opens up. STEP 2 STEP 3 Find and plot the vertex.

  4. The x-coordinate of the vertex isb , or1. – 2a EXAMPLE 2 Graph y = ax2 + b x + c To find the y-coordinate, substitute 1 for xin the function and simplify. y = 3(1)2– 6(1) + 2 =– 1 So, the vertex is(1, – 1). STEP 4 Plot two points. Choose two x-values less than the x-coordinate of the vertex. Then find the corresponding y-values.

  5. EXAMPLE 2 Standardized Test Practice STEP5 Reflect the points plotted in Step 4 in the axis of symmetry. STEP6 Draw a parabola through the plotted points.

  6. EXAMPLE 2 Graph y = ax2 + b x + c

  7. 1. Find the axis of symmetry and vertex of the graph of the function y= x2 – 2x – 3. ANSWER ANSWER x = 1, (1, –4). for Examples 1 and 2 GUIDED PRACTICE 2. Graph the function y = 3x2 + 12x – 1. Label the vertex and axis of symmetry.

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